Home Multiplicative topological indices of honeycomb derived networks
Article Open Access

Multiplicative topological indices of honeycomb derived networks

  • Jiang-Hua Tang , Mustafa Habib , Muhammad Younas , Muhammad Yousaf and Waqas Nazeer EMAIL logo
Published/Copyright: March 23, 2019
Download Article (PDF)
Open Physics
From the journal Open Physics Volume 17 Issue 1

Abstract

Topological indices are the numerical values associated with chemical structures that correlate physico-chemical properties with structural properties. There are various classes of topological indices such as degree based topological indices, distance based topological indices and counting related topological indices. Among these classes, degree based topological indices are of great importance and play a vital role in chemical graph theory, particularly in chemistry. In this report, we have computed the multiplicative degree based topological indices of honeycomb derived networks of dimensions I, 2, 3 and 4.

Keywords: Honeycomb network; topological index; degree; chemical graph theory
PACS: 81.05-t; 81.07.Nb

1 Introduction

Mathematical modeling of chemical reaction networks consists of a variety of methods for approaching questions about the dynamical behavior of chemical reactions arising in realworld applications. After the invention of the law of mass action, dynamical properties of reaction networks have been extensively studied in both chemistry and physics. The essential steps for this study were the introduction of the detailed equilibrium of complex chemical reactions by Rudolf Wegscheider [1], the development of the quantitative theory of chemical chain reactions by Nikolay Semyonov [2], the development of catalytic reactions by Cyril Norman Hinshelwood [3] and many other results.

Three epochs of chemical dynamics can be observed in the flow of research and publications [4]. These times can be associated with leaders: the first is the Van’t Hoff era, the second is the Semenov Hinshelwood era and the third is definitely the Aris era. The "times" can be distinguished on the basis of the priorities of scientific leaders:

  • Van’t Hoff looked for general laws of chemical reaction related to specific chemical properties. The term "chemical dynamics" belongs to van’t Hoff.

  • The goal of Hinshelwood Semenov, was to explore the critical phenomena observed in many chemical systems, especially in flames. The concept of chain reaction explored by these researchers influenced many sciences, particularly nuclear physics and engineering.

  • Aris’ activity was concentrated on the detailed systematization of mathematical ideas and approaches.

Rutherford Aris initiated the mathematical discipline called chemical reaction network theory. The work of R. Aris in the journal Archive for Rational Mechanics and Analysis has opened a series of works by other authors (informed by R. Aris). The most famous works of this series are the works of Frederick J. Krambeck [6], Roy Jackson, Friedrich Josef Maria Horn [7], Martin Feinberg [8] and others [9], which were published in the 1970s. In continuation of his early work in this area, R. Aris mentions the work of N.Z. Shapiro, L. S. Shapley [10], where a significant part of his scientific program was endorsed. Since then, a large number of researchers internationally [11, 12, 13, 14, 15, 16, 17, 18, 19, 20] have further developed the chemical reaction network theory.

The honeycomb and hexagonal networks have been known as crucial for evolutionary biology, in particular, for the evolution of cooperation, where overlapping triangles are vital for the propagation of cooperation in social dilemmas. For relevant research, see [21, 22]. In the hexagonal network HX (n), the parameter n is the number of vertices on each side of the network [23], whereas for the honeycomb network HC(n), n is the number of hexagons between a boundary and central hexagon [23]. Due to the significance of topological indices in chemistry, a lot of research has been done in this area. For further studies of topological indices of various graph families, see [24, 25, 26, 27, 28].

Let us consider a graph as shown in Figure 3.

Figure 1 Hexagonal network
Figure 1

Hexagonal network

Figure 2 Honeycomb network
Figure 2

Honeycomb network

Figure 3 Graph G
Figure 3

Graph G

The stellation of G is denoted by St(G) and can be obtained by adding a vertex in each face of G and then by join these vertices to all vertices of the respective face (see Figure 4).

Figure 4 Stellation of G (dotted)
Figure 4

Stellation of G (dotted)

The dual Du(G) of a graph G is a graph that has a vertex for each face of G. The graph has an edge whenever two faces of G are separated from each other by an edge, and a self-loop when the same face appears on both sides of an

edge, see Figure 5. Hence the number of faces of a graph is equal to the number of edges of its dual.

Figure 5 Dual of graph G (dotted)
Figure 5

Dual of graph G (dotted)

In the dual graph, if we delete the vertex corresponding to the bounded face of the planar graph, which is unique, we get the bounded dual Bdu(G) (see Figure 6).

Figure 6 Bounded dual of graph G (dotted)
Figure 6

Bounded dual of graph G (dotted)

Given a connected plane graph G, its medial graph M(G) has a vertex for each edge of G and an edge between two vertices for each face of G in which their corresponding edges occur consecutively (see Figure 7).

Figure 7 Medial of G (dotted)
Figure 7

Medial of G (dotted)

In this report, we aim to compute multiplicative degree-based topological indices of networks derived from honeycomb networks by taking stellation, dual, bounded dual, and medial graphs of the honeycomb network.

2 Topological indices

Amolecular graph is a simple graph in chemical graph theory in which atoms are represented by vertices and chemical bonds are represented by edges. A graph is connected if there is a connection between any pair of vertices. A network is a connected graph which has no multiple edge and no loop. The number of vertices which are connected to a fixed v vertex is called the degree of v and is denoted by dv . The distance between two vertices is the length of shortest path between them. The concept of valence in chemistry and concept of degree are somewhat closely related. For details on the basics of graph theory, refer to the book [29]. Quantitative structure-activity and structure-property relationships predict the properties and biological activities of materials. In these studies, topological indices and some physicochemical properties are used to predict bioactivity of chemical compounds [30, 31, 32, 33].

Throughout this paper, G denotes a connected graph, V and E denote the vertex set and the edge set and dv denotes the degree of a vertex. The topological index of the graph of a chemical compound is a number which can be used to characterize the represented chemical compound and help to predict its physicochemical properties.Wiener laid the foundation of topological index in 1947 .He approximated the boiling points of alkanes and introduced the Wiener index [34]. Up to now more than 140 topological indices have been defined but no single index is enough to determine all physicochemical properties; but these topological indices together can do this to some extent. Later, in 1975, Milan Randić introduced the Randić index, [35]. In 1998, Bollobas and Erdos [36] and Amic et al. [37] proposed the generalized Randić index which has been studied by both chemists and mathematicians [38]. The Randić index is one of the most popular, most studied and most applied topological indices. Many reviews, papers and books [39, 40, 41, 42, 43, 44] are written on this simple graph invariant. Some indices related to Wiener’s work are the first and second multiplicative Zagreb indices [45], respectively:

I I 1 G = ∏ u ∈ V G d u 2 I I 2 G = ∏ u v ∈ E G d u ⋅ d v

and the Narumi-Katayama index [46]:

N K G = ∏ u ∈ V G d u

Like the Wiener index, these types of indices are the focus of considerable research in computational chemistry [47, 48, 49, 50]. For example, in 2011 I. Gutman [47] characterized the multiplicative Zagreb indices for trees and determined the unique trees that obtained maximum and minimum values for M1(G) and M2(G), respectively. S. Wang and the last author [50] then extended Gutman’s result to the following index for k-trees:

W 1 s G = ∏ u ∈ V G d u s .

Notice that s = 1, 2 correspond to the Narumi-Katayama and Zagreb index, respectively. Based on the successful consideration of multiplicative Zagreb indices, M. Eliasi et al. [51] continued to define a new multiplicative version of the first Zagreb index as

I I 1 ∗ G = ∏ u v ∈ E G d u + d v .

Furthering the concept of indexing with the edge set, the first author introduced the first and second hyper-Zagreb indices of a graph [52]. They are defined as

H I I 1 G = ∏ u v ∈ E G d u + d v 2
H I I 2 G = ∏ u v ∈ E G d u ⋅ d v 2

In [53], Kulli et al. defined the first and second generalized Zagreb indices:

M Z 1 a G = ∏ u v ∈ E G d u + d v a
M Z 2 a G = ∏ u v ∈ E G d u ⋅ d v a

Multiplicative sum connectivity and multiplicative product connectivity indices [54] are define as

S C I I G = ∏ u v ∈ E G 1 d u + d v
P C I I G = ∏ u v ∈ E G 1 d u ⋅ d v

Multiplicative atomic bond connectivity index and multiplicative Geometric arithmetic index are defined as

A B C I I G = ∏ u v ∈ E G d u + d v − 2 d u ⋅ d v
G A I I G = ∏ u v ∈ E G 2 d u ⋅ d v d u + d v
G A a I I G = ∏ u v ∈ E G 2 d u ⋅ d v d u + d v a

3 Computational results

In this section we give our computational results.

Theorem 1

Let HcDN1 (n) be the honeycomb derived network of dimension 1. Then

  1. I I 2 H c D N 1 n = 6 42 5 30 18 6 × 15 12 × 30 18 × 36 27 n 3 57 n

  2. I I 1 ∗ H c D N 1 n = 2 30 × 3 36 11 18 9 6 × 8 12 × 11 18 × 12 27 n 12 57 n

  3. H I I 1 H c D N 1 n = 2 60 × 3 72 11 36 81 6 × 64 12 × 121 8 × 144 27 n 144 57 n

  4. H I I 2 H c D N 1 n = 6 84 5 60 5 60 × 6 108 n 2 180 × 3 144 n

  5. M Z 1 a H c D N 1 n = 2 30 a × 3 36 a 11 18 a 2 36 a × 3 12 a × 11 18 a × 12 27 a n 12 57 a n

  6. M Z 2 a H c D N 1 n = 6 42 a 5 30 a 3 42 a × 5 30 a × 2 24 a × 36 27 a n 36 57 a n

  7. X I I H c D N 1 n = 11 9 2 15 × 3 18 6 57 × 3 27 2 n 6 27 n × 2 18 × 3 93 2 × 11 9 n

  8. χ I I H c D N 1 n = 5 15 6 21 2 45 × 3 36 5 15 × 6 27 n n

  9. A B C I I H c D N 1 n = 5 30 2 6 × 3 45 2 45 2 × 3 60 × 7 3 × 10 27 2 n 5 87 2 × 6 27 n n

  10. G A I I H c D N 1 n = 11 18 2 3 × 6 15 2 12 × 3 9 × 5 15 11 18 n

  11. G A a I I H c D N 1 n = 11 18 a 3 15 a × 5 15 a × 2 3 a 3 9 a × 5 15 a × 2 12 a 11 18 a n

Proof:

The honeycomb derived network of dimension one HcDN1 (n) is obtained by taking the union of the honeycomb network and its stellation, which is a planar graph. In the honeycomb derived network HcDN1 (n),

V H c D N 1 n = 9 n 2 − 3 n + 1
E H c D N 1 n = 27 n 2 − 21 n + 6

There are five types of edges in E (HcDN1 (n)) based on the degree of end vertices, i.e.,

E 1 H c D N 1 n = u v ∈ E H c D N 1 n : d u = 3 , d v = 3 E 2 H c D N 1 n = u v ∈ E H c D N 1 n : d u = 3 , d v = 5 E 3 H c D N 1 n = u v ∈ E H c D N 1 n : d u = 3 , d v = 6 E 4 H c D N 1 n = u v ∈ E H c D N 1 n : d u = 5 , d v = 6 E 5 H c D N 1 n = u v ∈ E H c D N 1 n : d u = 6 , d v = 6

It can be observed from Figure 1 that

E 1 H c D N 1 n = 6 E 2 H c D N 1 n = 12 n − 1 E 3 H c D N 1 n = 6 n E 4 H c D N 1 n = 18 n − 1 E 5 H c D N 1 n = 27 n 2 − 57 n + 30

Now,

I I 2 H c D N 1 n = ∏ u v ∈ E H c D N 1 n d u ⋅ d v = 3 × 3 6 × 3 × 5 12 n − 1 × 3 × 6 6 n × 5 × 6 18 n − 1 × 6 × 6 27 n 2 − 57 n + 30 = 6 42 5 30 18 6 × 15 12 × 30 18 × 36 27 n 3 57 n .
I I 1 ∗ H c D N 1 n = ∏ u v ∈ E H c D N 1 n d u + d v = 3 + 3 6 × 3 + 5 12 n − 1 × 3 + 6 6 n × 5 + 6 18 n − 1 × 6 + 6 27 n 2 − 57 n + 30 = 2 30 × 3 36 11 18 9 6 × 8 12 × 11 18 × 12 27 n 12 57 n .
H I I 1 H c D N 1 n = ∏ u v ∈ E H c D N 1 n d u + d v 2 = 3 + 3 2 6 × 3 + 5 2 12 n − 1 × 3 + 6 2 6 n × 5 + 6 2 18 n − 1 × 6 + 6 2 27 n 2 − 57 n + 30 = 2 60 × 3 72 11 36 81 6 × 64 12 × 121 8 × 144 27 n 144 57 n .
H I I 2 H c D N 1 n = ∏ u v ∈ E H c D N 1 n d u ⋅ d v 2 = 3 × 3 2 6 × 3 × 5 2 12 n − 1 × 3 × 6 2 6 n × 5 × 6 2 18 n − 1 × 6 × 6 2 27 n 2 − 57 n + 30 = 6 84 5 60 5 60 × 6 108 n 2 180 × 3 144 n .
M Z 1 a H c D N 1 n = ∏ u v ∈ E H c D N 1 n d u + d v a = 3 + 3 a 6 × 3 + 5 a 12 n − 1 × 3 + 6 a 6 n × 5 + 6 a 18 n − 1 × 6 + 6 a 27 n 2 − 57 n + 30 = 2 30 a × 3 36 a 11 18 a 2 36 a × 3 12 a × 11 18 a × 12 27 a n 12 57 a n .

M Z 2 a H c D N 1 n = ∏ u v ∈ E H c D N 1 n d u ⋅ d v a = 3 × 3 a 6 × 3 × 5 a 12 n − 1 × 3 × 6 a 6 n × 5 × 6 a 18 n − 1 × 6 × 6 a 27 n 2 − 57 n + 30 = 6 42 a 5 30 a 3 42 a × 5 30 a × 2 24 a × 36 27 a n 36 57 a n .
X I I H c D N 1 n = ∏ u v ∈ E H c D N 1 n 1 d u + d v = 1 3 + 3 6
× 1 3 + 5 12 n − 1 × 1 3 + 6 6 n × 1 5 + 6 18 n − 1 × 1 6 + 6 27 n 2 − 57 n + 30 = 11 9 2 15 × 3 18 6 57 × 3 27 2 n 6 27 n × 2 18 × 3 93 2 × 11 9 n .
χ I I H c D N 1 n = ∏ u v ∈ E H c D N 1 n 1 d u ⋅ d v = 1 3 ⋅ 3 6 × 1 3 ⋅ 5 12 n − 1 × 1 3 ⋅ 6 6 n × 1 5 ⋅ 6 18 n − 1 × 1 6 ⋅ 6 27 n 2 − 57 n + 30 = 5 15 6 21 2 45 × 3 36 5 15 × 6 27 n n .
A B C I I H c D N 1 n = ∏ u v ∈ E H c D N 1 n d u + d v − 2 d u ⋅ d v = 3 + 3 − 2 3 ⋅ 3 6 × 3 + 5 − 2 3 ⋅ 5 12 n − 1 × 3 + 6 − 2 3 ⋅ 6 6 n × 5 + 6 − 2 5 ⋅ 6 18 n − 1 × 6 + 6 − 2 6 ⋅ 6 27 n 2 − 57 n + 30 = 5 30 2 6 × 3 45 2 45 2 × 3 60 × 7 3 × 10 27 2 n 5 87 2 × 6 27 n n .
G A I I H c D N 1 n = ∏ u v ∈ E H c D N 1 n 2 d u ⋅ d v d u + d v = 2 3 ⋅ 3 3 + 3 6 × 2 3 ⋅ 5 3 + 5 12 n − 1 × 2 3 ⋅ 6 3 + 6 6 n × 2 5 ⋅ 6 5 + 6 12 n − 1 × 2 6 ⋅ 6 6 + 6 27 n 2 − 57 n + 30 = 11 18 2 3 × 6 15 2 12 × 3 9 × 5 15 11 18 n .
G A a I I H c D N 1 n = ∏ u v ∈ E H c D N 1 n 2 d u ⋅ d v d u + d v a = 2 3 ⋅ 3 3 + 3 a 6 × 2 3 ⋅ 5 3 + 5 a 12 n − 1 × 2 3 ⋅ 6 3 + 6 a 6 n × 2 5 ⋅ 6 5 + 6 a 18 n − 1 × 2 6 ⋅ 6 6 + 6 a 27 n 2 − 57 n + 30 = 11 18 a 3 15 a × 5 15 a × 2 3 a 3 9 a × 5 15 a × 2 12 a 11 18 a n .

Theorem 2

Let HcDN 2 (n) be the honeycomb derived network of dimension 2. Then

  1. I I 2 H c D N 2 n = 2 84 × 3 228 5 150 2 108 n × 3 72 n × 5 90 2 428 × 3 162 n

  2. I I 1 ∗ H c D N 2 n = 3 138 × 7 18 × 19 12 2 42 × 5 30 × 11 30 × 13 12 2 63 n × 5 18 × 11 18 × 3 54 n × 13 6 2 87 × 3 18 n

  3. H I I 1 H c D N 2 n = 3 276 × 7 36 × 19 24 2 84 × 5 60 × 11 60 × 13 24 2 126 n × 11 36 × 3 108 n × 13 12 × 5 36 2 126 × 3 300 n

  4. H I I 2 H c D N 2 n = 2 168 × 3 456 5 300 2 216 n × 3 144 n × 5 180 2 456 × 3 324 n

  5. M Z 1 a H c D N 2 n = 3 97 × 7 18 × 11 30 × 19 12 2 188 × 5 30 × 13 2 a 2 63 n × 3 54 n × 5 18 × 11 18 2 63 × 3 150 a n

  6. M Z 2 a H c D N 2 n = 2 89 × 3 233 5 145 a 2 72 n × 3 72 n × 5 85 2 197 × 3 167 a n

  7. X I I H c D N 2 n = 2 21 × 5 15 × 11 15 × 13 6 3 69 × 7 9 × 19 6 2 32 × 3 151 2 5 9 × 11 9 × 13 3 × 2 31 n × 3 − 53 2 × 6 n n + 1 n

  8. χ I I H c D N 2 n = 5 75 12 54 × 3 114 2 114 × 3 81 5 45 × 2 54 n × 3 36 n n

  9. A B C I I H c D N 2 n = 2 30 × 3 102 × 5 75 × 11 9 × 17 6 × 19 3 13 6 × 7 18 2 18 × 3 99 × 7 9 × 13 6 × 2 18 n × 5 9 2 n × 11 9 2 n 5 99 2 × 11 27 2 × 3 36 n n

  10. G A I I H c D N 2 n = 2 96 × 11 30 × 13 12 3 24 × 5 45 × 7 18 × 19 12 2 12 × 3 69 × 5 27 11 18 × 13 6 × 2 72 n × 3 18 n n

  11. G A a I I H c D N 2 n = 2 96 × 11 30 × 13 12 3 24 × 5 45 × 7 18 × 19 12 a 2 12 × 3 69 × 5 27 11 18 × 13 6 × 2 72 n × 3 18 n a n

3.1 Proof:

The honeycomb derived network of dimension 2 HcDN 2 (n) is obtained by taking the union of the honeycomb network, its stellation and its bounded dual, which is a non-planar graph. In the honeycomb derived network HcDN2 (n),

V H c D N 2 n = 9 n 2 − 3 n + 1 E H c D N 2 n = 27 n 2 − 21 n + 6

There are sixteen types of edges in E (HcDN2 (n)) based on the degree of end vertices, i.e,

E 1 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 3 , d v = 3 E 2 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 3 , d v = 5 E 3 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 3 , d v = 9 E 4 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 3 , d v = 10 E 5 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 5 , d v = 6 E 6 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 5 , d v = 9 E 7 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 5 , d v = 10 E 8 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 6 , d v = 6 E 9 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 6 , d v = 9 E 10 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 6 , d v = 10 E 11 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 6 , d v = 12 E 12 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 9 , d v = 10 E 13 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 9 , d v = 12 E 14 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 10 , d v = 10 E 15 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 10 , d v = 12 E 16 H c D N 2 n = u v ∈ E H c D N 2 n : d u = 12 , d v = 12

It can be observed from Figure 2 that

E 1 H c D N 2 n = 6 E 2 H c D N 2 n = 12 n − 1 E 3 H c D N 2 n = 12 E 4 H c D N 2 n = 6 n − 2 E 5 H c D N 2 n = 6 n − 1 E 6 H c D N 2 n = 12 E 7 H c D N 2 n = 12 n − 2 E 8 H c D N 2 n = 9 n 2 − 21 n + 12 E 9 H c D N 2 n = 12 E 10 H c D N 2 n = 18 n − 2 E 11 H c D N 2 n = 18 n 2 − 54 n + 42 E 12 H c D N 2 n = 12 E 13 H c D N 2 n = 6 E 14 H c D N 2 n = 6 n − 3 E 15 H c D N 2 n = 12 n − 2 E 16 H c D N 2 n = 9 n 2 − 33 n + 30

Now,

I I 2 H c D N 2 n = ∏ u v ∈ E H c D N 2 n d u ⋅ d v = 3 × 3 6 × 3 × 5 12 n − 1 × 3 × 9 12 × 3 × 10 6 n − 2 × 5 × 6 6 n − 1 × 5 × 9 12 × 5 × 10 12 n − 2 × 6 × 6 9 n 2 − 21 n + 12 × 6 × 9 12 × 6 × 10 18 n − 2 × 6 × 12 18 n 2 − 54 n + 42 × 9 × 10 12 × 9 × 12 6 × 10 × 10 6 n − 3 × 10 × 12 12 n − 2 × 12 × 12 9 n 2 − 33 n + 30 = 2 84 × 3 228 5 150 2 108 n × 3 72 n × 5 90 2 428 × 3 162 n .
I I 1 ∗ H c D N 2 n = ∏ u v ∈ E H c D N 2 n d u + d v = 3 + 3 6 × 3 + 5 12 n − 1 × 3 + 9 12 × 3 + 10 6 n − 2 × 5 + 6 6 n − 1 × 5 + 9 12 × 5 + 10 12 n − 2 × 6 + 6 9 n 2 − 21 n + 12 × 6 + 9 12 × 6 + 10 18 n − 2 × 6 + 12 18 n 2 − 54 n + 42 × 9 + 10 12 × 9 + 12 6 × 10 + 10 6 n − 3 × 10 + 12 12 n − 2 × 12 + 12 9 n 2 − 33 n + 30 = 3 138 × 7 18 × 19 12 2 42 × 5 30 × 11 30 × 13 12 2 63 n × 5 18 × 11 18 × 3 54 n × 13 6 2 87 × 3 18 n .
H I I 1 H c D N 2 n = ∏ u v ∈ E H c D N 2 n d u + d v 2 = 3 + 3 2 6 × 3 + 5 2 12 n − 1 × 3 + 9 2 12 × 3 + 10 2 6 n − 2 × 5 + 6 2 6 n − 1 × 5 + 9 2 12 × 5 + 10 2 12 n − 2 × 6 + 6 2 9 n 2 − 21 n + 12 × 6 + 9 2 12 × 6 + 10 2 18 n − 2 × 6 + 12 2 18 n 2 − 54 n + 42 × 9 + 10 2 12 × 9 + 12 2 6 × 10 + 10 2 6 n − 3 × 10 + 12 2 12 n − 2 × 10 + 12 2 9 n 2 − 33 n + 30 = 3 276 × 7 36 × 19 24 2 84 × 5 60 × 11 60 × 13 24 2 126 n × 11 36 × 3 108 n × 13 12 × 5 36 2 126 × 3 300 n .
H I I 2 H c D N 2 n = ∏ u v ∈ E H c D N 2 n d u ⋅ d v 2 = 3 × 3 2 6 × 3 × 5 2 12 n − 1 × 3 × 9 2 12 × 3 × 10 2 6 n − 2 × 5 × 6 2 6 n − 1 × 5 × 9 2 12 × 5 × 10 2 12 n − 2 × 6 × 6 2 9 n 2 − 21 n + 12 × 6 × 9 2 12 × 6 × 10 2 18 n − 2 × 6 × 12 2 18 n 2 − 54 n + 42 × 9 × 10 2 12 × 9 × 12 2 6 × 10 × 10 2 6 n − 3 × 10 × 12 2 12 n − 2 × 12 × 12 2 9 n 2 − 33 n + 30 = 2 168 × 3 456 5 300 2 216 n × 3 144 n × 5 180 2 456 × 3 324 n .
M Z 1 a H c D N 2 n = ∏ u v ∈ E H c D N 2 n d u + d v a = 3 + 3 a 6 × 3 + 5 a 12 n − 1 × 3 + 9 a 12 × 3 + 10 a 6 n − 2 × 5 + 6 a 6 n − 1 × 5 + 9 a 12 × 5 + 10 a 12 n − 2 × 6 + 6 a 9 n 2 − 21 n + 12 × 6 + 9 a 12 × 6 + 10 a 18 n − 2 × 6 + 12 a 18 n 2 − 54 n + 42 × 9 + 10 a 12 × 9 + 12 a 6 × 10 + 10 a 6 n − 3 × 10 + 12 a 12 n − 2 × 10 + 12 a 9 n 2 − 33 n + 30 = 3 97 × 7 18 × 11 30 × 19 12 2 188 × 5 30 × 13 2 a 2 63 n × 3 54 n × 5 18 × 11 18 2 63 × 3 150 a n .
M Z 2 a H c D N 2 n = ∏ u v ∈ E H c D N 2 n d u ⋅ d v a = 3 × 3 a 6 × 3 × 5 a 12 n − 1 × 3 × 9 a 12 × 3 × 10 a 6 n − 2 × 5 × 6 a 6 n − 1 × 5 × 9 a 12 × 5 × 10 a 12 n − 2 × 6 × 6 a 9 n 2 − 21 n + 12 × 6 × 9 a 12 × 6 × 10 a 18 n − 2 × 6 × 12 a 18 n 2 − 54 n + 42 × 9 × 10 a 12 × 9 × 12 a 6 × 10 × 10 a 6 n − 3 × 10 × 12 a 12 n − 2 × 12 × 12 a 9 n 2 − 33 n + 30 = 2 89 × 3 233 5 145 a 2 72 n × 3 72 n × 5 85 2 197 × 3 167 a n .
X I I H c D N 2 n = ∏ u v ∈ E H c D N 2 n 1 d u + d v = 1 3 + 3 6 × 1 3 + 5 12 n − 1 × 1 3 + 9 12 × 1 3 + 10 6 n − 2 × 1 5 + 6 6 n − 1 × 1 5 + 9 12 × 1 5 + 10 12 n − 2 × 1 6 + 6 9 n 2 − 21 n + 12 × 1 6 + 9 12 × 1 6 + 10 18 n − 2 × 1 6 + 12 18 n 2 − 54 n + 42 × 1 9 + 10 12 × 1 9 + 12 6 × 1 10 + 10 6 n − 3 × 1 10 + 12 12 n − 2 × 1 12 + 12 9 n 2 − 33 n + 30 = 2 21 × 5 15 × 11 15 × 13 6 3 69 × 7 9 × 19 6 2 32 × 3 151 2 5 9 × 11 9 × 13 3 × 2 31 n × 3 − 53 2 × 6 n n + 1 n .
χ I I H c D N 2 n = ∏ u v ∈ E H c D N 2 n 1 d u . d v = 1 3.3 6 × 1 3.5 12 n − 1 × 1 3.9 12 × 1 3.10 6 n − 2 × 1 5.6 6 n − 1 × 1 5.9 12 × 1 5.10 12 n − 2 × 1 6.6 9 n 2 − 21 n + 12 × 1 6 + 9 12 × 1 6 + 10 18 n − 2 × 1 6 + 12 18 n 2 − 54 n + 42 × 1 9 + 10 12 × 1 9 + 12 6 × 1 10 + 10 6 n − 3 × 1 10 + 12 12 n − 2 × 1 12 + 12 9 n 2 − 33 n + 30 = 5 75 12 54 × 3 114 2 114 × 3 81 5 45 × 2 54 n × 3 36 n n .
A B C I I H c D N 2 n = ∏ u v ∈ E H c D N 2 n d u + d v − 2 d u ⋅ d v = 3 + 3 − 2 3 ⋅ 3 6 × 3 + 5 − 2 3 ⋅ 5 12 n − 1 × 3 + 9 − 2 3 ⋅ 9 12 × 3 + 10 − 2 3 ⋅ 10 6 n − 2 × 5 + 6 − 2 5 ⋅ 6 6 n − 1 × 5 + 9 − 2 5 ⋅ 9 12 × 5 + 10 − 2 5 ⋅ 10 12 n − 2 × 6 + 6 − 2 6 ⋅ 6 9 n 2 − 21 n + 12 × 6 + 9 − 2 6 ⋅ 9 12 × 6 + 10 − 2 6 ⋅ 10 18 n − 2 × 6 + 12 − 2 6 ⋅ 12 18 n 2 − 54 n + 42 × 9 + 10 − 2 9 ⋅ 10 12 × 9 + 12 − 2 9 ⋅ 12 6 × 10 + 10 − 2 10 ⋅ 10 6 n − 3 × 10 + 12 − 2 10 ⋅ 12 12 n − 2 × 12 + 12 − 2 12 ⋅ 12 9 n 2 − 33 n + 30 = 2 30 × 3 102 × 5 75 × 11 9 × 17 6 × 19 3 13 6 × 7 18 2 18 × 3 99 × 7 9 × 13 6 × 2 18 n × 5 9 2 n × 11 9 2 n 5 99 2 × 11 27 2 × 3 36 n n .
G A I I H c D N 2 n = ∏ u v ∈ E H c D N 2 n 2 d u ⋅ d v d u + d v = 2 3 ⋅ 3 3 + 3 6 × 2 3 ⋅ 5 3 + 5 12 n − 1 × 2 3 ⋅ 9 3 + 9 12 × 2 3 ⋅ 10 3 + 10 6 n − 2 × 2 5 ⋅ 6 5 + 6 6 n − 1 × 2 5 ⋅ 9 5 + 9 12 × 2 5 ⋅ 10 5 + 10 12 n − 2 × 2 6 ⋅ 6 6 + 6 9 n 2 − 21 n + 12 × 2 6 ⋅ 9 6 + 9 12 × 2 6 ⋅ 10 6 + 10 18 n − 2 × 2 6 ⋅ 12 6 + 12 18 n 2 − 54 n + 42 × 2 9 ⋅ 10 9 + 10 12 × 2 9 ⋅ 12 9 + 12 6 × 2 10 ⋅ 10 10 + 10 6 n − 3 × 2 10 ⋅ 12 10 + 12 12 n − 2 × 2 12 ⋅ 12 12 + 12 9 n 2 − 33 n + 30 = 2 96 × 11 30 × 13 12 3 24 × 5 45 × 7 18 × 19 12 2 12 × 3 69 × 5 27 11 18 × 13 6 × 2 72 n × 3 18 n n .
G A α I I H c D N 2 n = ∏ u v ∈ E H c D N 2 n 2 d u ⋅ d v d u + d v α = 2 3 ⋅ 3 3 + 3 a 6 × 2 3 ⋅ 5 3 + 5 a 12 n − 1 × 2 3 ⋅ 9 3 + 9 a 12 × 2 3 ⋅ 10 3 + 10 a 6 n − 2 × 2 5 ⋅ 6 5 + 6 a 6 n − 1 × 2 5 ⋅ 9 5 + 9 a 12 × 2 5 ⋅ 10 5 + 10 a 12 n − 2 × 2 6 ⋅ 6 6 + 6 a 9 n 2 − 21 n + 12 × 2 6 ⋅ 9 6 + 9 a 12 × 2 6 ⋅ 10 6 + 10 a 18 n − 2 × 2 6 ⋅ 12 6 + 12 a 18 n 2 − 54 n + 42 × 2 9 ⋅ 10 9 + 10 a 12 × 2 9 ⋅ 12 9 + 12 a 6 × 2 10 ⋅ 10 10 + 10 a 6 n − 3 × 2 10 ⋅ 12 10 + 12 a 12 n − 2 × 2 12 ⋅ 12 12 + 12 a 9 n 2 − 33 n + 30 = 2 96 × 11 30 × 13 12 3 24 × 5 45 × 7 18 × 19 12 a 2 12 × 3 69 × 5 27 11 18 × 13 6 × 2 72 n × 3 18 n a n .

Theorem 3

Let HcDN3 (n) be the honeycomb derived network of dimension 3. Then

  1. I I 2 H c D N 3 n = 2 30 × 3 78 5 30 2 108 n × 3 108 n × 5 30 3 84 × 3 162 n

  2. I I 1 ∗ H c D N 3 n = 2 96 × 3 30 5 12 × 11 18 2 108 n × 3 54 n × 5 12 × 7 12 × 11 18 2 186 × 3 72 n

  3. H I I 1 H c D N 3 n = 2 192 × 3 60 5 24 × 11 36 2 216 n × 3 108 n × 5 24 × 7 24 × 11 36 2 372 × 2 144 n

  4. H I I 2 H c D N 3 n = 2 60 × 3 156 5 60 2 216 n × 3 216 n × 5 60 2 168 × 3 324 n

  5. M Z 1 a H c D N 3 n = 2 96 a × 3 30 a 5 12 a × 11 18 a 2 108 n × 3 54 n × 5 12 × 7 12 × 11 18 2 186 × 3 72 a n

  6. M Z 2 a H c D N 3 n = 2 30 a × 3 78 a 5 30 a 2 108 n × 3 108 n × 5 30 3 84 × 3 162 a n

  7. X I I H c D N 3 n = 5 6 × 11 9 2 48 × 3 15 2 93 × 3 36 2 54 n × 3 27 n × 5 6 × 7 6 × 11 9 n

  8. χ I I H c D N 3 n = 5 15 2 15 × 3 39 2 42 × 3 81 2 54 n × 3 54 n × 5 15 n

  9. A B C I I H c D N 3 n = 5 42 2 6 × 3 57 × 7 6 2 27 n × 2 12 × 3 105 × 7 6 2 27 n × 3 54 n × 5 63 n

  10. G A I I H c D N 3 n = 3 9 × 11 18 2 69 × 5 3 2 102 × 5 3 3 9 × 7 12 × 11 18 n

  11. G A a I I H c D N 3 n = 3 9 a × 11 18 a 2 69 a × 5 3 a 2 102 × 5 3 3 9 × 7 12 × 11 18 a n

Proof

The honeycomb derived network of dimension 3 HcDN 3 (n) is obtained by taking the union of the honeycomb network, its stellation and its medial, which is a non-planar graph.

In the honeycomb derived network HcDN3 (n),

V H c D N 3 n = 18 n 2 − 6 n + 1 E H c D N 3 n = 54 n 2 − 42 n + 12

There are seven types of edges in E (HcDN3 (n)) based on the degree of end vertices, i.e.,

E 1 H c D N 3 n = u v ∈ E H c D N 3 n : d u = 3 , d v = 4 E 2 H c D N 3 n = u v ∈ E H c D N 3 n : d u = 3 , d v = 6 E 3 H c D N 3 n = u v ∈ E H c D N 3 n : d u = 4 , d v = 4 E 4 H c D N 3 n = u v ∈ E H c D N 3 n : d u = 4 , d v = 5 E 5 H c D N 3 n = u v ∈ E H c D N 3 n : d u = 4 , d v = 6 E 6 H c D N 3 n = u v ∈ E H c D N 3 n : d u = 5 , d v = 6 E 7 H c D N 3 n = u v ∈ E H c D N 3 n : d u = 6 , d v = 6
E 1 H c D N 3 n = 12 n E 2 H c D N 3 n = 6 n E 3 H c D N 3 n = 6 n E 4 H c D N 3 n = 12 n − 1 E 5 H c D N 3 n = 12 n − 1 E 6 H c D N 3 n = 18 n − 1 E 7 H c D N 3 n = 54 n 2 − 108 n + 54

Now,

I I 2 H c D N 3 n = ∏ u v ∈ E H c D N 3 n d u ⋅ d v = 3 × 4 12 n × 3 × 6 6 n × 4 × 4 6 n × 4 × 5 12 n − 1 × 4 × 6 12 n − 1 × 5 × 6 18 n − 1 × 6 × 6 54 n 2 − 108 n + 54 = 2 30 × 3 78 5 30 2 108 n × 3 108 n × 5 30 3 84 × 3 162 n .
I I 1 ∗ H c D N 3 n = ∏ u v ∈ E H c D N 3 n d u + d v = 3 + 4 12 n × 3 + 6 6 n × 4 + 4 6 n × 4 + 5 12 n − 1 × 4 + 6 12 n − 1 × 5 + 6 18 n − 1 × 6 + 6 54 n 2 − 108 n + 54 = 2 96 × 3 30 5 12 × 11 18 2 108 n × 3 54 n × 5 12 × 7 12 × 11 18 2 186 × 3 72 n .
H I I 1 H c D N 3 n = ∏ u v ∈ E H c D N 3 n d u + d v 2 = 3 + 4 2 12 n × 3 + 6 2 6 n × 4 + 4 2 6 n × 4 + 5 2 12 n − 1 × 4 + 6 2 12 n − 1 × 5 + 6 2 18 n − 1 × 6 + 6 2 54 n 2 − 108 n + 54 = 2 192 × 3 60 5 24 × 11 36 2 216 n × 3 108 n × 5 24 × 7 24 × 11 36 2 372 × 2 144 n .
H I I 2 H c D N 3 n = ∏ u v ∈ E H c D N 3 n d u ⋅ d v 2 = 3 × 4 2 12 n × 3 × 6 2 6 n × 4 × 4 2 6 n × 4 × 5 2 12 n − 1 × 4 × 6 2 12 n − 1 × 5 × 6 2 18 n − 1 × 6 × 6 2 54 n 2 − 108 n + 54 = 2 60 × 3 156 5 60 2 216 n × 3 216 n × 5 60 2 168 × 3 324 n .
M Z 1 a H c D N 3 n = ∏ u v ∈ E H c D N 3 n d u + d v a = 3 + 4 a 12 n × 3 + 6 a 6 n × 4 + 4 a 6 n × 4 + 5 a 12 n − 1 × 4 + 6 a 12 n − 1 × 5 + 6 a 18 n − 1 × 6 + 6 a 54 n 2 − 108 n + 54 = 2 96 a × 3 30 a 5 12 a × 11 18 a 2 108 n × 3 54 n × 5 12 × 7 12 × 11 18 2 186 × 3 72 a n .
X I I H c D N 3 n = ∏ u v ∈ E H c D N 3 n 1 d u + d v = 1 3 + 4 12 n × 1 3 + 6 6 n × 1 4 + 4 6 n × 1 4 + 5 12 n − 1 × 1 4 + 6 12 n − 1 × 1 5 + 6 18 n − 1 × 1 6 + 6 54 n 2 − 108 n + 54 = 5 6 × 11 9 2 48 × 3 15 2 93 × 3 36 2 54 n × 3 27 n × 5 6 × 7 6 × 11 9 n .
χ I I H c D N 3 n = ∏ u v ∈ E H c D N 3 n 1 d u ⋅ d v = 1 3 ⋅ 4 12 n × 1 3 ⋅ 6 6 n × 1 4 ⋅ 4 6 n × 1 4 ⋅ 5 12 n − 1 × 1 4 ⋅ 6 12 n − 1 × 1 5 ⋅ 6 18 n − 1 × 1 6 ⋅ 6 54 n 2 − 108 n + 54 = 5 15 2 15 × 3 39 2 42 × 3 81 2 54 n × 3 54 n × 5 15 n .
A B C I I H c D N 3 n = ∏ u v ∈ E H c D N 3 n d u + d v − 2 d u ⋅ d v = 3 + 4 − 2 3 ⋅ 4 12 n × 3 + 6 − 2 3 ⋅ 6 6 n × 4 + 4 − 2 4 ⋅ 4 6 n × 4 + 5 − 2 4 ⋅ 5 12 n − 1 × 4 + 6 − 2 4 ⋅ 6 12 n − 1 × 5 + 6 − 2 5 ⋅ 6 18 n − 1 × 6 + 6 − 2 6 ⋅ 6 54 n 2 − 108 n + 54 = 5 42 2 6 × 3 57 × 7 6 2 27 n × 2 12 × 3 105 × 7 6 2 27 n × 3 54 n × 5 63 n .
G A I I H c D N 3 n = ∏ u v ∈ E H c D N 3 n 2 d u ⋅ d v d u + d v = 2 3 ⋅ 4 3 + 4 12 n × 2 3 ⋅ 6 3 + 6 6 n × 2 4 ⋅ 4 4 + 4 6 n × 2 4 ⋅ 5 4 + 5 12 n − 1 × 2 4 ⋅ 6 4 + 6 12 n − 1 × 2 5 ⋅ 6 5 + 6 18 n − 1 × 2 6 ⋅ 6 6 + 6 54 n 2 − 108 n + 54 = 3 9 × 11 18 2 69 × 5 3 2 102 × 5 3 3 9 × 7 12 × 11 18 n .
G A a I I H c D N 3 n = ∏ u v ∈ E H c D N 3 n 2 d u ⋅ d v d u + d v a = 2 3 ⋅ 4 3 + 4 a 12 n × 2 3 ⋅ 6 3 + 6 a 6 n × 2 4 ⋅ 4 4 + 4 a 6 n × 2 4 ⋅ 5 4 + 5 a 12 n − 1 × 2 4 ⋅ 6 4 + 6 a 12 n − 1 × 2 5 ⋅ 6 5 + 6 a 18 n − 1 × 2 6 ⋅ 6 6 + 6 a 54 n 2 − 108 n + 54 = 3 9 a × 11 18 a 2 69 a × 5 3 a 2 102 × 5 3 3 9 × 7 12 × 11 18 a n .

Theorem 4

Let HcDN4 (n) be the honeycomb derived network of dimension 4. Then

  1. I I 1 ∗ H c D N 4 n = 3 156 × 19 12 2 276 × 5 18 × 7 282 × 11 30 × 13 12 2 99 n × 3 45 n × 5 18 × 7 156 × 11 18 × 13 6 2 9 × 3 153 n

  2. H I I 1 H c D N 4 n = 3 312 × 19 24 2 552 × 5 36 × 7 564 × 11 60 × 13 24 2 198 n × 3 90 n × 5 36 × 7 312 × 11 36 × 13 12 2 18 × 3 306 n

  3. H I I 2 H c D N 4 n = 2 192 3 672 × 5 804 2 324 n × 3 252 n × 3 72 × 5 420 2 468 n

  4. M Z 2 a H c D N 4 n = 2 96 3 336 × 5 402 a 2 162 n × 3 162 n × 3 36 × 5 210 2 234 a n

  5. X I I H c D N 4 n = 2 138 × 5 9 × 7 141 × 11 15 × 13 6 3 78 × 19 6 2 9 2 × 3 153 2 2 99 2 n × 3 45 2 n × 5 9 × 7 78 × 11 9 × 13 3 n

  6. χ I I H c D N 4 n = 3 168 × 5 201 2 48 2 117 2 81 n × 3 63 n × 3 18 × 5 105 n

  7. A B C I I H c D N 4 n = 5 213 × 11 9 × 13 6 × 17 6 × 19 3 2 282 × 7 24 2 483 2 n × 3 57 n × 5 36 n × 2 399 2 × 3 273 × 5 123 × 7 15 × 11 15 2 11 33 2 n

  8. G A I I H c D N 4 n = 2 30 × 11 30 × 13 12 5 33 × 7 18 × 19 12 2 27 n × 2 3 × 3 51 × 5 15 3 18 n × 7 12 × 11 18 × 13 6 n

  9. G A a I I H c D N 4 n = 2 30 × 11 30 × 13 12 5 33 × 7 18 × 19 12 a 2 27 n × 2 3 × 3 51 × 5 15 3 18 n × 7 12 × 11 18 × 13 6 a n

Proof

The honeycomb derived network of dimension 4 HcDN 4 (n) is obtained by taking the union of the honeycomb network, its stellation, its bounded dual and its medial, which is a non-planar graph.

In the honeycomb derived network HcDN4 (n),

V H c D N 4 n = 18 n 2 − 6 n + 1 E H c D N 4 n = 63 n 2 − 57 n + 18

There are sixteen types of edges in E (HcDN4 (n)) based on the degree of end vertices, i.e.,

E 1 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 3 , d v = 4 E 2 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 3 , d v = 9 E 3 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 3 , d v = 10 E 4 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 4 , d v = 4 E 5 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 4 , d v = 5 E 6 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 4 , d v = 6 E 7 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 5 , d v = 6 E 8 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 5 , d v = 9 E 9 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 5 , d v = 10 E 10 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 6 , d v = 6 E 11 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 6 , d v = 9 E 12 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 6 , d v = 10 E 13 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 6 , d v = 12 E 14 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 9 , d v = 10 E 15 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 9 , d v = 12 E 16 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 10 , d v = 10 E 17 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 10 , d v = 12 E 18 H c D N 4 n = u v ∈ E H c D N 4 n : d u = 12 , d v = 12
E 1 H c D N 4 n = 12 n E 2 H c D N 4 n = 12 E 3 H c D N 4 n = 6 n − 2 E 4 H c D N 4 n = 6 n E 5 H c D N 4 n = 12 n − 1 E 6 H c D N 4 n = 12 n − 1 E 7 H c D N 4 n = 6 n − 1 E 8 H c D N 4 n = 12 E 9 H c D N 4 n = 12 n − 2 E 10 H c D N 4 n = 36 n 2 − 72 n + 36 E 11 H c D N 4 n = 12 E 12 H c D N 4 n = 18 n − 2 E 13 H c D N 4 n = 18 n 2 − 54 n + 42 E 14 H c D N 4 n = 12 E 15 H c D N 4 n = 6 E 16 H c D N 4 n = 6 n − 3 E 17 H c D N 4 n = 12 n − 2 E 18 H c D N 4 n = 9 n 2 − 33 n + 30

Now,

I I 2 H c D N 4 n = ∏ u v ∈ E H c D N 4 n d u ⋅ d v = 3 × 4 12 n × 3 × 9 12 × 3 × 10 6 n − 2 × 4 × 4 6 n × 4 × 5 12 n − 1 × 4 × 6 12 n − 1 × 5 × 6 6 n − 1 × 5 × 9 12 × 5 × 10 12 n − 2 × 6 × 6 36 n 2 − 72 n + 36 × 6 × 9 12 × 6 × 10 18 n − 2 × 6 × 12 18 n 2 − 54 n + 42 × 9 × 10 12 × 9 × 12 6 × 10 × 10 6 n − 3 × 10 × 12 12 n − 2 × 12 × 12 9 n 2 − 33 n + 30 = 2 96 3 336 × 5 402 2 162 n × 3 162 n × 3 36 × 5 210 2 234 n .
I I 1 ∗ H c D N 4 n = ∏ u v ∈ E H c D N 4 n d u + d v = 3 + 4 12 n × 3 + 9 12 × 3 + 10 6 n − 2 × 4 + 4 6 n × 4 + 5 12 n − 1 × 4 + 6 12 n − 1 × 5 + 6 6 n − 1 × 5 + 9 12 × 5 + 10 12 n − 2 × 6 + 6 36 n 2 − 72 n + 36 × 6 + 9 12 × 6 + 10 18 n − 2 × 6 + 12 18 n 2 − 54 n + 42 × 9 + 10 12 × 9 + 12 6 × 10 + 10 6 n − 3 × 10 + 12 12 n − 2 × 12 + 12 9 n 2 − 33 n + 30 = 3 156 × 19 12 2 276 × 5 18 × 7 282 × 11 30 × 13 12 2 99 n × 3 45 n × 5 18 × 7 156 × 11 18 × 13 6 2 9 × 3 153 n .
H I I 1 H c D N 4 n = ∏ u v ∈ E H c D N 4 n d u + d v 2 = 3 + 4 2 12 n × 3 + 9 2 12 × 3 + 10 2 6 n − 2 × 4 + 4 2 6 n × 4 + 5 2 12 n − 1 × 4 + 6 2 12 n − 1 × 5 + 6 2 6 n − 1 × 5 + 9 2 12 × 5 + 10 2 12 n − 2 × 6 + 6 2 36 n 2 − 72 n + 36 × 6 + 9 2 12 × 6 + 10 2 18 n − 2 × 6 + 12 2 18 n 2 − 54 n + 42 × 9 + 10 2 12 × 9 + 12 2 6 × 10 + 10 2 6 n − 3 × 10 + 12 2 12 n − 2 × 10 + 12 2 9 n 2 − 33 n + 30 = 3 312 × 19 24 2 552 × 5 36 × 7 564 × 11 60 × 13 24 2 198 n × 3 90 n × 5 36 × 7 312 × 11 36 × 13 12 2 18 × 3 306 n .
H I I 2 H c D N 4 n = ∏ u v ∈ E H c D N 4 n d u ⋅ d v 2 = 3 × 4 2 12 n × 3 × 9 2 12 × 3 × 10 2 6 n − 2 × 4 × 4 2 6 n × 4 × 5 2 12 n − 1 × 4 × 6 2 12 n − 1 × 5 × 6 2 6 n − 1 × 5 × 9 2 12 × 5 × 10 2 12 n − 2 × 6 × 6 2 36 n 2 − 72 n + 36 × 6 × 9 2 12 × 6 × 10 2 18 n − 2 × 6 × 12 2 18 n 2 − 54 n + 42 × 9 × 10 2 12 × 9 × 12 2 6 × 10 × 10 2 6 n − 3 × 10 × 12 2 12 n − 2 × 12 × 12 2 9 n 2 − 33 n + 30 = 2 192 3 672 × 5 804 2 324 n × 3 252 n × 3 72 × 5 420 2 468 n .
M Z 1 a H c D N 4 n = ∏ u v ∈ E H c D N 4 n d u + d v a = 3 + 4 a 12 n × 3 + 9 a 12 × 3 + 10 a 6 n − 2 × 4 + 4 a 6 n × 4 + 5 a 12 n − 1 × 4 + 6 a 12 n − 1 × 5 + 6 a 6 n − 1 × 5 + 9 a 12 × 5 + 10 a 12 n − 2 × 6 + 6 a 36 n 2 − 72 n + 36 × 6 + 9 a 12 × 6 + 10 a 18 n − 2 × 6 + 12 a 18 n 2 − 54 n + 42 × 9 + 10 a 12 × 9 + 12 a 6 × 10 + 10 a 6 n − 3 × 10 + 12 a 12 n − 2 × 10 + 12 a 9 n 2 − 33 n + 30 = 3 156 × 19 12 2 276 × 5 18 × 7 282 × 11 30 × 13 12 a 2 99 n × 3 45 n × 5 18 × 7 156 × 11 18 × 13 6 2 9 × 3 153 a n .
M Z 2 a H c D N 4 n = ∏ u v ∈ E H c D N 4 n d u ⋅ d v a = 3 × 3 a 6 × 3 × 5 a 12 n − 1 × 3 × 9 a 12 × 3 × 10 a 6 n − 2 × 5 × 6 a 6 n − 1 × 5 × 9 a 12 × 5 × 10 a 12 n − 2 × 6 × 6 a 9 n 2 − 21 n + 12 × 6 × 9 a 12 × 6 × 10 a 18 n − 2 × 6 × 12 a 18 n 2 − 54 n + 42 × 9 × 10 a 12 × 9 × 12 a 6 × 10 × 10 a 6 n − 3 × 10 × 12 a 12 n − 2 × 12 × 12 a 9 n 2 − 33 n + 30 = 2 96 3 336 × 5 402 a 2 162 n × 3 162 n × 3 36 × 5 210 2 234 a n .
X I I H c D N 4 n = ∏ u v ∈ E H c D N 4 n 1 d u + d v = 1 3 + 4 12 n × 1 3 + 9 12 × 1 3 + 10 6 n − 2 × 1 4 + 4 6 n × 1 4 + 5 12 n − 1 × 1 4 + 6 12 n − 1 × 1 5 + 6 6 n − 1 × 1 5 + 9 12 × 1 5 + 10 12 n − 2 × 1 6 + 6 36 n 2 − 72 n + 36 × 1 6 + 9 12 × 1 6 + 10 18 n − 2 × 1 6 + 12 18 n 2 − 54 n + 42 × 1 9 + 10 12 × 1 9 + 12 6 × 1 10 + 10 6 n − 3 × 1 10 + 12 12 n − 2 × 1 12 + 12 9 n 2 − 33 n + 30 = 2 138 × 5 9 × 7 141 × 11 15 × 13 6 3 78 × 19 6 2 9 2 × 3 153 2 2 99 2 n × 3 45 2 n × 5 9 × 7 78 × 11 9 × 13 3 n .
χ I I H c D N 4 n = ∏ u v ∈ E H c D N 4 n 1 d u . d v 1 3.4 12 n × 1 3.9 12 1 3.10 6 ( n − 2 ) × 1 4.4 6 n × 1 4.5 12 ( n − 1 ) × 1 4.6 12 ( n − 1 ) × 1 5.6 6 ( n − 1 ) × 1 5.9 12 × 1 5.10 12 ( n − 2 ) × 1 6.6 36 n 2 − 72 n + 36 × 1 6.9 12 × 1 6.10 18 ( n − 2 ) × 1 6.12 18 n 2 − 54 n + 42 × 1 9.10 12 × 1 9.12 6 × 1 10.10 6 ( n − 3 ) × 1 10.12 12 ( n − 2 ) × 1 12.12 9 n 2 − 33 n + 30 = 3 168 × 5 201 2 48 2 117 2 81 n × 3 63 n × 3 18 × 5 105
A B C I I ( H c D N 4 ( n ) ) = ∏ u v ∈ E H c D N 4 n d u + d v − 2 d u ⋅ d v = 3 + 4 − 2 3 ⋅ 4 12 n × 3 + 9 − 2 3 ⋅ 9 12 × 3 + 10 − 2 3 ⋅ 10 6 n − 2 × 4 + 4 − 2 4 ⋅ 4 6 n × 4 + 5 − 2 4 ⋅ 5 12 n − 1 × 4 + 4 − 2 4 ⋅ 6 12 n − 1 × 5 + 6 − 2 5 ⋅ 6 6 n − 1 × 5 + 9 − 2 5 ⋅ 9 12 × 5 + 10 − 2 5 ⋅ 10 12 n − 2 × 6 + 6 − 2 6 ⋅ 6 36 n 2 − 72 n + 36 × 6 + 9 − 2 6 ⋅ 9 12 × 6 + 10 − 2 6 ⋅ 10 18 n − 2 × 6 + 12 − 2 6 ⋅ 12 18 n 2 − 54 n + 42 × 9 + 10 − 2 9 ⋅ 10 12 × 9 + 12 − 2 9 ⋅ 12 6 × 10 + 10 − 2 10 ⋅ 10 6 n − 3 × 10 + 12 − 2 10 ⋅ 12 12 n − 2 × 12 + 12 − 2 12 ⋅ 12 9 n 2 − 33 n + 30 = 5 213 × 11 9 × 13 6 × 17 6 × 19 3 2 282 × 7 24 2 483 2 n × 3 57 n × 5 36 n × 2 399 2 × 3 273 × 5 123 × 7 15 × 11 15 2 11 33 2 n .
G A I I H c D N 4 n = ∏ u v ∈ E H c D N 4 n 2 d u ⋅ d v d u + d v = 2 3 ⋅ 4 3 + 4 12 n × 2 3 ⋅ 9 3 + 9 12 × 2 3 ⋅ 10 3 + 10 6 n − 2 × 2 4 ⋅ 4 4 + 4 6 n × 2 4 ⋅ 5 4 + 5 12 n − 1 × 2 4 ⋅ 6 4 + 6 12 n − 1 × 2 5 ⋅ 6 5 + 6 6 n − 1 × 2 5 ⋅ 9 5 + 9 12 × 2 5 ⋅ 10 5 + 10 12 n − 2 × 2 6 ⋅ 6 6 + 6 36 n 2 − 72 n + 36 × 2 6 ⋅ 9 6 + 9 12 × 2 6 ⋅ 10 6 + 10 18 n − 2 × 2 6 ⋅ 12 6 + 12 18 n 2 − 54 n + 42 × 2 9 ⋅ 10 9 + 10 12 × 2 9 ⋅ 12 9 + 12 6 × 2 10 ⋅ 10 10 + 10 6 n − 3 × 2 10 ⋅ 12 10 + 12 12 n − 2 × 2 12 ⋅ 12 12 + 12 9 n 2 − 33 n + 30 = 2 30 × 11 30 × 13 12 5 33 × 7 18 × 19 12 2 27 n × 2 3 × 3 51 × 5 15 3 18 n × 7 12 × 11 18 × 13 6 n .
G A a I I H c D N 4 n = ∏ u v ∈ E H c D N 4 n 2 d u ⋅ d v d u + d v a = 2 3 ⋅ 4 3 + 4 a 12 n × 2 3 ⋅ 9 3 + 9 a 12 × 2 3 ⋅ 10 3 + 10 a 6 n − 2 × 2 4 ⋅ 4 4 + 4 a 6 n × 2 4 ⋅ 5 4 + 5 a 12 n − 1 × 2 4 ⋅ 6 4 + 6 a 12 n − 1 × 2 5 ⋅ 6 5 + 6 a 6 n − 1 × 2 5 ⋅ 9 5 + 9 a 12 × 2 5 ⋅ 10 5 + 10 a 12 n − 2 × 2 6 ⋅ 6 6 + 6 a 36 n 2 − 72 n + 36 × 2 6 ⋅ 9 6 + 9 a 12 × 2 6 ⋅ 10 6 + 10 a 18 n − 2 × 2 6 ⋅ 12 6 + 12 a 18 n 2 − 54 n + 42 × 2 9 ⋅ 10 9 + 10 a 12 × 2 9 ⋅ 12 9 + 12 a 6 × 2 10 ⋅ 10 10 + 10 a 6 n − 3 × 2 10 ⋅ 12 10 + 12 a 12 n − 2 × 2 12 ⋅ 12 12 + 12 a 9 n 2 − 33 n + 30 = 2 30 × 11 30 × 13 12 5 33 × 7 18 × 19 12 a 2 27 n × 2 3 × 3 51 × 5 15 3 18 n × 7 12 × 11 18 × 13 6 a n .

4 Conclusions

In recent years, the computational physics problem on special molecular network structures has gained attention in theoretical physics. From the standpoint of graph theory, one learns a real-valued function that assigns scores to molecular graphs. What is important is the relative physicochemical characters of the corresponding compounds induced by those scores. This topic is distinct from both classical graph theory and physical computing problems, and it is natural to ask what kinds topological index and molecular structure generalizations hold for this problem. Although there have been several recent advances in developing results for various settings of the molecular calculation problem, the study of topological indices of special molecular networks has been largely limited to the special mathematical setting. In this manuscript, the authors study the multiplicative degree based topological indices of honeycomb derived networks of dimension 1, 2, 3 and 4, and their specific expressions are obtained. These results can also play a vital part in the determination of the significance of honeycomb derived networks. For example, it has been experimentally verified that the first Zagreb index is directly related to total π-electron energy. The Randić index is useful for determining physicochemical properties of alkanes as noticed by the chemist Melan Randić in 1975. He noticed the correlation between the Randic index R and several physicochemical properties of alkanes like enthalpies of formation, boiling points, chromatographic retention times, vapor pressure and surface areas. Calculations of distance-based and surface-based topological indices of these networks are an open problem in this area of research. Also, the construction of new networks by taking line graphs and para-line graphs of honeycomb networks is an interesting problem.

Acknowledgement

Many thanks to both reviewers for their valuable comments that helped us to improve the quality of this paper. This research is supported by Project Supported by the Natural Science Fund Project of Anhui Xinhua University (Grant Nos.2017zr011).

References

[1] Wegscheider R., Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme, Monatshefte für Chemie und verwandte Teile anderer Wissenschaften, 1911, 32, 849-906.10.1007/BF01517735Search in Google Scholar

[2] Semenov N.N., Some problems relating to chain reactions and to the theory of combustion, Nobel Lecture, 1956, 487-514.Search in Google Scholar

[3] Hinshelwood C., Chemical kinetics in the past few decades, Science, 1957, 125(3250), 679-682.10.1126/science.125.3250.679Search in Google Scholar

[4] Gorban A.N., Yablonsky G.S., Three waves of chemical dynamics, Math. Model. Natur. Phenom., 2015, 10(5), 1-5.10.1051/mmnp/201510501Search in Google Scholar

[5] Aris R., Prolegomena to the rational analysis of systems of chemical reactions, Archive Ration, Mech. Anal., 1965, 19(2), pp.81-99.10.1007/BF00282276Search in Google Scholar

[6] Krambeck F.J., The mathematical structure of chemical kinet ics in homogeneous single-phase systems, Archive Ration, Mech. Anal., 1970, 38(5), 317-347.10.1007/BF00251527Search in Google Scholar

[7] Horn F., Jackson R., Generalmass action kinetics. Archive for rational mechanics and analysis, 1972, 47(2), 81-116.10.1007/BF00251225Search in Google Scholar

[8] Feinberg M., Complex balancing in general kinetic systems, Archive Ration, Mech. Anal., 1972, 49(3), 187-194.10.1007/BF00255665Search in Google Scholar

[9] Aris R., Prolegomena to the rational analysis of systems of chemical reactions II. Some addenda, Archive Ration, Mech. Anal., 1968, 27(5), 356-364.10.1016/S1874-5970(99)80014-9Search in Google Scholar

[10] Shapiro N.Z., Shapley L.S.,Mass action laws and the Gibbs free energy function, J. Soc. Industr. Appl. Math., 1965, 13(2), 353-375.10.1137/0113020Search in Google Scholar

[11] Érdi P., Tóth J.,Mathematical models of chemical reactions: theory and applications of deterministic and stochastic models, Manchester University Press, 1989.Search in Google Scholar

[12] Kunze H., Siegel D., Monotonicity properties of chemical reactions with a single initial bimolecular step, J. Math. Chem., 2002, 31(4), 339-344.10.1023/A:1021064420348Search in Google Scholar

[13] Mincheva M., Siegel D., Nonnegativity and positiveness of so lutions to mass action reaction–diffusion systems, J. Math. Chem., 2007, 42(4), 1135-1145.10.1007/s10910-007-9292-0Search in Google Scholar

[14] De Leenheer P., Angeli D., Sontag E.D., Monotone chemical reaction networks, J. Math. Chem., 2007, 41(3), pp.295-314.10.1007/s10910-006-9075-zSearch in Google Scholar

[15] Banaji M., Donnell P., Baigent S., Pmatrix properties, injectivity, and stability in chemical reaction systems, SIAM J. Appl. Math., 2007, 67(6), 1523-1547.10.1137/060673412Search in Google Scholar

[16] Craciun G., Pantea C., Identifiability of chemical reaction networks, J. Math. Chem., 2008, 44(1), 244-259.10.1007/s10910-007-9307-xSearch in Google Scholar

[17] Domijan M., Kirkilionis M., Bistability and oscillations in chemical reaction networks, J. Math. Biol., 2009, 59(4), 467-501.10.1007/s00285-008-0234-7Search in Google Scholar PubMed

[18] Gorban A.N., Yablonsky G.S., Extended detailed balance for systems with irreversible reactions, Chem. Eng. Sci., 2011, 66(21), 5388-5399.10.1016/j.ces.2011.07.054Search in Google Scholar

[19] Feliu E., Knudsen M., Wiuf C., Signaling cascades: consequences of varying substrate and phosphatase levels, Adv. Syst. Biol., 2012, 81-94.10.1007/978-1-4419-7210-1_4Search in Google Scholar PubMed

[20] Otero-Muras I., Banga J.R., Alonso A.A., Characterizing multistationarity regimes in biochemical reaction networks, PLoS One, 2012, 7(7), e39194.10.1371/journal.pone.0039194Search in Google Scholar PubMed PubMed Central

[21] Imran M., Baig A.Q., Ali H. Rehman S.U., On topological properties of poly honeycomb networks, Period. Math. Hungarica, 2016, 73(1), 100-119.10.1007/s10998-016-0132-5Search in Google Scholar

[22] Hayat S.,Malik M.A. Imran M., Computing topological indices of honey-comb derived networks, Romanian J. Inf. Sci. Techn., 2015, 18, 144-165.Search in Google Scholar

[23] Ahmad M.S., Nazeer W., Kang S.M., Imran M., Gao W., Calculating degree-based topological indices of dominating David derived networks, Open Phys., 15(1), 1015-1021.10.1515/phys-2017-0126Search in Google Scholar

[24] Abdo H., Dimitrov D., GaoW., On the irregularity of some molecular structures, Canadian J. Chem., 2016, 95(2), 174- 183.10.1139/cjc-2016-0539Search in Google Scholar

[25] Ahmadi M.B., Dimitrov D., Gutman I., Hosseini S.A., Disproving a conjecture on trees with minimal atom-bond connectivity index, MATCH Commun. Math. Comput. Chem, 2014, 72(3), 685-698.Search in Google Scholar

[26] Dimitrov D.,Onstructural properties of treeswith minimal atom-bond connectivity index II: Bounds on B1-and B2-branches, Discr. Appl. Math., 2016,204, 90-116.10.1016/j.dam.2015.10.010Search in Google Scholar

[27] Dimitrov D., Du, Z., da Fonseca, C.M., On structural properties of trees with minimal atom-bond connectivity index III: Trees with pendent paths of length three, Appl. Math. Comput., 2016, 282, 276-290.10.1016/j.amc.2016.02.019Search in Google Scholar

[28] Guirao J.L.G., de Bustos M.T., Dynamics of pseudo-radioactive chemical products via sampling theory, J. Math. Chem., 2012, 50(2), 374-378.10.1007/s10910-010-9788-xSearch in Google Scholar

[29] West D.B., Introduction to graph theory (Vol. 2), Upper Saddle River: Prentice hall, 2001.Search in Google Scholar

[30] Gao W., Wang Y., Wang W., Shi L., The first multiplication atom-bond connectivity index of molecular structures in drugs, Saudi Pharm. J., 2017, 25(4), 548-555.10.1016/j.jsps.2017.04.021Search in Google Scholar

[31] Gao W., Wang Y., Basavanagoud B., Jamil M.K., Characteristics studies ofmolecular structures in drugs, Saudi Pharmaceu- tical Journal, 2017, 25(4), pp.580-586.10.1016/j.jsps.2017.04.027Search in Google Scholar

[32] Gao W., Wang W., Dimitrov D., Wang, Y., Nano Properties Analysis via Fourth Multiplicative ABC Indicator Calculating, Arabian J. Chem., 2018, 11(6), 793-801.10.1016/j.arabjc.2017.12.024Search in Google Scholar

[33] Deutsch E., Klavzar S., M-Polynomial and degree-based topological indices, Iranian J. Math. Chem., 2015, 6, 93-102Search in Google Scholar

[34] Wiener H., Structural determination of parafln boiling points, J. Amer. Chem. Soc., 1947, 69(1), 17-20.10.1021/ja01193a005Search in Google Scholar

[35] Randić M., Characterization of molecular branching, J. Amer. Chem. Soc., 1975, 97(23), 6609-6615.10.1021/ja00856a001Search in Google Scholar

[36] Bollobás B., Erdös P., Graphs of extremal weights, Ars Combinatoria, 1998, 50, 225-233.10.1016/S0012-365X(98)00320-3Search in Google Scholar

[37] Amić D., Bešlo D., Lucˇić, B., Nikolić S., Trinajstić, N., The vertex connectivity index revisited, J. Chem. Inf. Comp. Sci., 1998, 38(5), 819-822.10.1021/ci980039bSearch in Google Scholar

[38] Hu Y., Li X., Shi Y., Xu T., Gutman I., On molecular graphs with smallest and greatest zeroth-order general Randic index, MATCH Commun. Math. Comput. Chem, 2005, 54(2), 425-434.Search in Google Scholar

[39] Li X., Gutman, I., Randić M., Mathematical aspects of Randićtype molecular structure descriptors, University, Faculty of Science, 2006.Search in Google Scholar

[40] Randić M.,Onhistory of the Randić index and emerging hostility toward chemical graph theory, MATCH Commun. Math. Comput. Chem, 2008, 59, 5-124.Search in Google Scholar

[41] Randić M., The connectivity index 25 years after, J. Mol. Graph. Model., 2001, 20(1), 19-35.10.1016/S1093-3263(01)00098-5Search in Google Scholar

[42] Gutman I., Furtula, B. eds., Recent results in the theory of Randić index. University, Faculty of Science.Search in Google Scholar

[43] Li X., Shi Y., A survey on the Randic index, MATCH Commun. Math. Comput. Chem, 2008, 59(1), 127-156.Search in Google Scholar

[44] Gutman I., Recent Results in the Theory of Randić Index, 2008, Kragujevac University.Search in Google Scholar

[45] Gutman I., Ruscić B., Trinajstić N., Wilcox Jr C.F., Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys., 1975, 62(9), 3399-3405.10.1063/1.430994Search in Google Scholar

[46] Narumi H., Katayama M., Simple topological index: A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons, Memoirs of the Faculty of Engineering, Hokkaido University, 1984, 16(3), 209-214.Search in Google Scholar

[47] Gutman I., Multiplicative Zagreb indices of trees, Bull. Soc. Math. Banja Luka, 2011, 18, 17-23.Search in Google Scholar

[48] Todeschini R., Ballabio D., Consonni V., Novel molecular descriptors based on functions of new vertex degrees, Math. Chem. Monographs, 2010, 73-100.Search in Google Scholar

[49] Todeschini, R., Consonni V., New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem, 2010, 64(2), 359-372.Search in Google Scholar

[50] Wang S., Wei B., Multiplicative Zagreb indices of k-trees, Discrete Applied Mathematics, 2015, 180,168-175.10.1016/j.dam.2014.08.017Search in Google Scholar

[51] Eliasi M., Iranmanesh A., Gutman I., Multiplicative versions of first Zagreb index, MATCH Commun. Math. Comp. Chem., 2012, 68(1), 217.Search in Google Scholar

[52] Kulli, V.R., Multiplicative hyper-Zagreb indices and coindices of graphs: computing these indices of some nanostructures, International Res. J. Pure Algebra, 2016, 6(7), 342-347.Search in Google Scholar

[53] Kulli V.R., Stone B.,Wang S., Wei, B., Generalised multiplicative indices of polycyclic aromatic hydrocarbons and benzenoid systems, Zeitschrift für Naturforschung A, 2017, 72(6), 573-576.10.1515/zna-2017-0104Search in Google Scholar

[54] Kulli V.R., Multiplicative connectivity indices of TUC4C8 [m, n] and TUC4 [m, n] nanotubes, J.Comp.Math. Sci., 2016, 7(11), 599-605.10.22457/ijfma.v11n1a6Search in Google Scholar
Received: 2018-02-24
Accepted: 2019-01-08
Published Online: 2019-03-23

© 2019 J.-H. Tang et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. Non-equilibrium Phase Transitions in 2D Small-World Networks: Competing Dynamics
  3. Harmonic waves solution in dual-phase-lag magneto-thermoelasticity
  4. Multiplicative topological indices of honeycomb derived networks
  5. Zagreb Polynomials and redefined Zagreb indices of nanostar dendrimers
  6. Solar concentrators manufacture and automation
  7. Idea of multi cohesive areas - foundation, current status and perspective
  8. Derivation method of numerous dynamics in the Special Theory of Relativity
  9. An application of Nwogu’s Boussinesq model to analyze the head-on collision process between hydroelastic solitary waves
  10. Competing Risks Model with Partially Step-Stress Accelerate Life Tests in Analyses Lifetime Chen Data under Type-II Censoring Scheme
  11. Group velocity mismatch at ultrashort electromagnetic pulse propagation in nonlinear metamaterials
  12. Investigating the impact of dissolved natural gas on the flow characteristics of multicomponent fluid in pipelines
  13. Analysis of impact load on tubing and shock absorption during perforating
  14. Energy characteristics of a nonlinear layer at resonant frequencies of wave scattering and generation
  15. Ion charge separation with new generation of nuclear emulsion films
  16. On the influence of water on fragmentation of the amino acid L-threonine
  17. Formulation of heat conduction and thermal conductivity of metals
  18. Displacement Reliability Analysis of Submerged Multi-body Structure’s Floating Body for Connection Gaps
  19. Deposits of iron oxides in the human globus pallidus
  20. Integrability, exact solutions and nonlinear dynamics of a nonisospectral integral-differential system
  21. Bounds for partition dimension of M-wheels
  22. Visual Analysis of Cylindrically Polarized Light Beams’ Focal Characteristics by Path Integral
  23. Analysis of repulsive central universal force field on solar and galactic dynamics
  24. Solitary Wave Solution of Nonlinear PDEs Arising in Mathematical Physics
  25. Understanding quantum mechanics: a review and synthesis in precise language
  26. Plane Wave Reflection in a Compressible Half Space with Initial Stress
  27. Evaluation of the realism of a full-color reflection H2 analog hologram recorded on ultra-fine-grain silver-halide material
  28. Graph cutting and its application to biological data
  29. Time fractional modified KdV-type equations: Lie symmetries, exact solutions and conservation laws
  30. Exact solutions of equal-width equation and its conservation laws
  31. MHD and Slip Effect on Two-immiscible Third Grade Fluid on Thin Film Flow over a Vertical Moving Belt
  32. Vibration Analysis of a Three-Layered FGM Cylindrical Shell Including the Effect Of Ring Support
  33. Hybrid censoring samples in assessment the lifetime performance index of Chen distributed products
  34. Study on the law of coal resistivity variation in the process of gas adsorption/desorption
  35. Mapping of Lineament Structures from Aeromagnetic and Landsat Data Over Ankpa Area of Lower Benue Trough, Nigeria
  36. Beta Generalized Exponentiated Frechet Distribution with Applications
  37. INS/gravity gradient aided navigation based on gravitation field particle filter
  38. Electrodynamics in Euclidean Space Time Geometries
  39. Dynamics and Wear Analysis of Hydraulic Turbines in Solid-liquid Two-phase Flow
  40. On Numerical Solution Of The Time Fractional Advection-Diffusion Equation Involving Atangana-Baleanu-Caputo Derivative
  41. New Complex Solutions to the Nonlinear Electrical Transmission Line Model
  42. The effects of quantum spectrum of 4 + n-dimensional water around a DNA on pure water in four dimensional universe
  43. Quantum Phase Estimation Algorithm for Finding Polynomial Roots
  44. Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia
  45. The Errors Recognition and Compensation for the Numerical Control Machine Tools Based on Laser Testing Technology
  46. Evaluation and Decision Making of Organization Quality Specific Immunity Based on MGDM-IPLAO Method
  47. Key Frame Extraction of Multi-Resolution Remote Sensing Images Under Quality Constraint
  48. Influences of Contact Force towards Dressing Contiguous Sense of Linen Clothing
  49. Modeling and optimization of urban rail transit scheduling with adaptive fruit fly optimization algorithm
  50. The pseudo-limit problem existing in electromagnetic radiation transmission and its mathematical physics principle analysis
  51. Chaos synchronization of fractional–order discrete–time systems with different dimensions using two scaling matrices
  52. Stress Characteristics and Overload Failure Analysis of Cemented Sand and Gravel Dam in Naheng Reservoir
  53. A Big Data Analysis Method Based on Modified Collaborative Filtering Recommendation Algorithms
  54. Semi-supervised Classification Based Mixed Sampling for Imbalanced Data
  55. The Influence of Trading Volume, Market Trend, and Monetary Policy on Characteristics of the Chinese Stock Exchange: An Econophysics Perspective
  56. Estimation of sand water content using GPR combined time-frequency analysis in the Ordos Basin, China
  57. Special Issue Applications of Nonlinear Dynamics
  58. Discrete approximate iterative method for fuzzy investment portfolio based on transaction cost threshold constraint
  59. Multi-objective performance optimization of ORC cycle based on improved ant colony algorithm
  60. Information retrieval algorithm of industrial cluster based on vector space
  61. Parametric model updating with frequency and MAC combined objective function of port crane structure based on operational modal analysis
  62. Evacuation simulation of different flow ratios in low-density state
  63. A pointer location algorithm for computer visionbased automatic reading recognition of pointer gauges
  64. A cloud computing separation model based on information flow
  65. Optimizing model and algorithm for railway freight loading problem
  66. Denoising data acquisition algorithm for array pixelated CdZnTe nuclear detector
  67. Radiation effects of nuclear physics rays on hepatoma cells
  68. Special issue: XXVth Symposium on Electromagnetic Phenomena in Nonlinear Circuits (EPNC2018)
  69. A study on numerical integration methods for rendering atmospheric scattering phenomenon
  70. Wave propagation time optimization for geodesic distances calculation using the Heat Method
  71. Analysis of electricity generation efficiency in photovoltaic building systems made of HIT-IBC cells for multi-family residential buildings
  72. A structural quality evaluation model for three-dimensional simulations
  73. WiFi Electromagnetic Field Modelling for Indoor Localization
  74. Modeling Human Pupil Dilation to Decouple the Pupillary Light Reflex
  75. Principal Component Analysis based on data characteristics for dimensionality reduction of ECG recordings in arrhythmia classification
  76. Blinking Extraction in Eye gaze System for Stereoscopy Movies
  77. Optimization of screen-space directional occlusion algorithms
  78. Heuristic based real-time hybrid rendering with the use of rasterization and ray tracing method
  79. Review of muscle modelling methods from the point of view of motion biomechanics with particular emphasis on the shoulder
  80. The use of segmented-shifted grain-oriented sheets in magnetic circuits of small AC motors
  81. High Temperature Permanent Magnet Synchronous Machine Analysis of Thermal Field
  82. Inverse approach for concentrated winding surface permanent magnet synchronous machines noiseless design
  83. An enameled wire with a semi-conductive layer: A solution for a better distibution of the voltage stresses in motor windings
  84. High temperature machines: topologies and preliminary design
  85. Aging monitoring of electrical machines using winding high frequency equivalent circuits
  86. Design of inorganic coils for high temperature electrical machines
  87. A New Concept for Deeper Integration of Converters and Drives in Electrical Machines: Simulation and Experimental Investigations
  88. Special Issue on Energetic Materials and Processes
  89. Investigations into the mechanisms of electrohydrodynamic instability in free surface electrospinning
  90. Effect of Pressure Distribution on the Energy Dissipation of Lap Joints under Equal Pre-tension Force
  91. Research on microstructure and forming mechanism of TiC/1Cr12Ni3Mo2V composite based on laser solid forming
  92. Crystallization of Nano-TiO2 Films based on Glass Fiber Fabric Substrate and Its Impact on Catalytic Performance
  93. Effect of Adding Rare Earth Elements Er and Gd on the Corrosion Residual Strength of Magnesium Alloy
  94. Closed-die Forging Technology and Numerical Simulation of Aluminum Alloy Connecting Rod
  95. Numerical Simulation and Experimental Research on Material Parameters Solution and Shape Control of Sandwich Panels with Aluminum Honeycomb
  96. Research and Analysis of the Effect of Heat Treatment on Damping Properties of Ductile Iron
  97. Effect of austenitising heat treatment on microstructure and properties of a nitrogen bearing martensitic stainless steel
  98. Special Issue on Fundamental Physics of Thermal Transports and Energy Conversions
  99. Numerical simulation of welding distortions in large structures with a simplified engineering approach
  100. Investigation on the effect of electrode tip on formation of metal droplets and temperature profile in a vibrating electrode electroslag remelting process
  101. Effect of North Wall Materials on the Thermal Environment in Chinese Solar Greenhouse (Part A: Experimental Researches)
  102. Three-dimensional optimal design of a cooled turbine considering the coolant-requirement change
  103. Theoretical analysis of particle size re-distribution due to Ostwald ripening in the fuel cell catalyst layer
  104. Effect of phase change materials on heat dissipation of a multiple heat source system
  105. Wetting properties and performance of modified composite collectors in a membrane-based wet electrostatic precipitator
  106. Implementation of the Semi Empirical Kinetic Soot Model Within Chemistry Tabulation Framework for Efficient Emissions Predictions in Diesel Engines
  107. Comparison and analyses of two thermal performance evaluation models for a public building
  108. A Novel Evaluation Method For Particle Deposition Measurement
  109. Effect of the two-phase hybrid mode of effervescent atomizer on the atomization characteristics
  110. Erratum
  111. Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance
  112. Erratum to: Energy converting layers for thin-film flexible photovoltaic structures
https://doi.org/10.1515/phys-2019-0003
Keywords for this article
Honeycomb network; topological index; degree; chemical graph theory
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Non-equilibrium Phase Transitions in 2D Small-World Networks: Competing Dynamics
  3. Harmonic waves solution in dual-phase-lag magneto-thermoelasticity
  4. Multiplicative topological indices of honeycomb derived networks
  5. Zagreb Polynomials and redefined Zagreb indices of nanostar dendrimers
  6. Solar concentrators manufacture and automation
  7. Idea of multi cohesive areas - foundation, current status and perspective
  8. Derivation method of numerous dynamics in the Special Theory of Relativity
  9. An application of Nwogu’s Boussinesq model to analyze the head-on collision process between hydroelastic solitary waves
  10. Competing Risks Model with Partially Step-Stress Accelerate Life Tests in Analyses Lifetime Chen Data under Type-II Censoring Scheme
  11. Group velocity mismatch at ultrashort electromagnetic pulse propagation in nonlinear metamaterials
  12. Investigating the impact of dissolved natural gas on the flow characteristics of multicomponent fluid in pipelines
  13. Analysis of impact load on tubing and shock absorption during perforating
  14. Energy characteristics of a nonlinear layer at resonant frequencies of wave scattering and generation
  15. Ion charge separation with new generation of nuclear emulsion films
  16. On the influence of water on fragmentation of the amino acid L-threonine
  17. Formulation of heat conduction and thermal conductivity of metals
  18. Displacement Reliability Analysis of Submerged Multi-body Structure’s Floating Body for Connection Gaps
  19. Deposits of iron oxides in the human globus pallidus
  20. Integrability, exact solutions and nonlinear dynamics of a nonisospectral integral-differential system
  21. Bounds for partition dimension of M-wheels
  22. Visual Analysis of Cylindrically Polarized Light Beams’ Focal Characteristics by Path Integral
  23. Analysis of repulsive central universal force field on solar and galactic dynamics
  24. Solitary Wave Solution of Nonlinear PDEs Arising in Mathematical Physics
  25. Understanding quantum mechanics: a review and synthesis in precise language
  26. Plane Wave Reflection in a Compressible Half Space with Initial Stress
  27. Evaluation of the realism of a full-color reflection H2 analog hologram recorded on ultra-fine-grain silver-halide material
  28. Graph cutting and its application to biological data
  29. Time fractional modified KdV-type equations: Lie symmetries, exact solutions and conservation laws
  30. Exact solutions of equal-width equation and its conservation laws
  31. MHD and Slip Effect on Two-immiscible Third Grade Fluid on Thin Film Flow over a Vertical Moving Belt
  32. Vibration Analysis of a Three-Layered FGM Cylindrical Shell Including the Effect Of Ring Support
  33. Hybrid censoring samples in assessment the lifetime performance index of Chen distributed products
  34. Study on the law of coal resistivity variation in the process of gas adsorption/desorption
  35. Mapping of Lineament Structures from Aeromagnetic and Landsat Data Over Ankpa Area of Lower Benue Trough, Nigeria
  36. Beta Generalized Exponentiated Frechet Distribution with Applications
  37. INS/gravity gradient aided navigation based on gravitation field particle filter
  38. Electrodynamics in Euclidean Space Time Geometries
  39. Dynamics and Wear Analysis of Hydraulic Turbines in Solid-liquid Two-phase Flow
  40. On Numerical Solution Of The Time Fractional Advection-Diffusion Equation Involving Atangana-Baleanu-Caputo Derivative
  41. New Complex Solutions to the Nonlinear Electrical Transmission Line Model
  42. The effects of quantum spectrum of 4 + n-dimensional water around a DNA on pure water in four dimensional universe
  43. Quantum Phase Estimation Algorithm for Finding Polynomial Roots
  44. Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia
  45. The Errors Recognition and Compensation for the Numerical Control Machine Tools Based on Laser Testing Technology
  46. Evaluation and Decision Making of Organization Quality Specific Immunity Based on MGDM-IPLAO Method
  47. Key Frame Extraction of Multi-Resolution Remote Sensing Images Under Quality Constraint
  48. Influences of Contact Force towards Dressing Contiguous Sense of Linen Clothing
  49. Modeling and optimization of urban rail transit scheduling with adaptive fruit fly optimization algorithm
  50. The pseudo-limit problem existing in electromagnetic radiation transmission and its mathematical physics principle analysis
  51. Chaos synchronization of fractional–order discrete–time systems with different dimensions using two scaling matrices
  52. Stress Characteristics and Overload Failure Analysis of Cemented Sand and Gravel Dam in Naheng Reservoir
  53. A Big Data Analysis Method Based on Modified Collaborative Filtering Recommendation Algorithms
  54. Semi-supervised Classification Based Mixed Sampling for Imbalanced Data
  55. The Influence of Trading Volume, Market Trend, and Monetary Policy on Characteristics of the Chinese Stock Exchange: An Econophysics Perspective
  56. Estimation of sand water content using GPR combined time-frequency analysis in the Ordos Basin, China
  57. Special Issue Applications of Nonlinear Dynamics
  58. Discrete approximate iterative method for fuzzy investment portfolio based on transaction cost threshold constraint
  59. Multi-objective performance optimization of ORC cycle based on improved ant colony algorithm
  60. Information retrieval algorithm of industrial cluster based on vector space
  61. Parametric model updating with frequency and MAC combined objective function of port crane structure based on operational modal analysis
  62. Evacuation simulation of different flow ratios in low-density state
  63. A pointer location algorithm for computer visionbased automatic reading recognition of pointer gauges
  64. A cloud computing separation model based on information flow
  65. Optimizing model and algorithm for railway freight loading problem
  66. Denoising data acquisition algorithm for array pixelated CdZnTe nuclear detector
  67. Radiation effects of nuclear physics rays on hepatoma cells
  68. Special issue: XXVth Symposium on Electromagnetic Phenomena in Nonlinear Circuits (EPNC2018)
  69. A study on numerical integration methods for rendering atmospheric scattering phenomenon
  70. Wave propagation time optimization for geodesic distances calculation using the Heat Method
  71. Analysis of electricity generation efficiency in photovoltaic building systems made of HIT-IBC cells for multi-family residential buildings
  72. A structural quality evaluation model for three-dimensional simulations
  73. WiFi Electromagnetic Field Modelling for Indoor Localization
  74. Modeling Human Pupil Dilation to Decouple the Pupillary Light Reflex
  75. Principal Component Analysis based on data characteristics for dimensionality reduction of ECG recordings in arrhythmia classification
  76. Blinking Extraction in Eye gaze System for Stereoscopy Movies
  77. Optimization of screen-space directional occlusion algorithms
  78. Heuristic based real-time hybrid rendering with the use of rasterization and ray tracing method
  79. Review of muscle modelling methods from the point of view of motion biomechanics with particular emphasis on the shoulder
  80. The use of segmented-shifted grain-oriented sheets in magnetic circuits of small AC motors
  81. High Temperature Permanent Magnet Synchronous Machine Analysis of Thermal Field
  82. Inverse approach for concentrated winding surface permanent magnet synchronous machines noiseless design
  83. An enameled wire with a semi-conductive layer: A solution for a better distibution of the voltage stresses in motor windings
  84. High temperature machines: topologies and preliminary design
  85. Aging monitoring of electrical machines using winding high frequency equivalent circuits
  86. Design of inorganic coils for high temperature electrical machines
  87. A New Concept for Deeper Integration of Converters and Drives in Electrical Machines: Simulation and Experimental Investigations
  88. Special Issue on Energetic Materials and Processes
  89. Investigations into the mechanisms of electrohydrodynamic instability in free surface electrospinning
  90. Effect of Pressure Distribution on the Energy Dissipation of Lap Joints under Equal Pre-tension Force
  91. Research on microstructure and forming mechanism of TiC/1Cr12Ni3Mo2V composite based on laser solid forming
  92. Crystallization of Nano-TiO2 Films based on Glass Fiber Fabric Substrate and Its Impact on Catalytic Performance
  93. Effect of Adding Rare Earth Elements Er and Gd on the Corrosion Residual Strength of Magnesium Alloy
  94. Closed-die Forging Technology and Numerical Simulation of Aluminum Alloy Connecting Rod
  95. Numerical Simulation and Experimental Research on Material Parameters Solution and Shape Control of Sandwich Panels with Aluminum Honeycomb
  96. Research and Analysis of the Effect of Heat Treatment on Damping Properties of Ductile Iron
  97. Effect of austenitising heat treatment on microstructure and properties of a nitrogen bearing martensitic stainless steel
  98. Special Issue on Fundamental Physics of Thermal Transports and Energy Conversions
  99. Numerical simulation of welding distortions in large structures with a simplified engineering approach
  100. Investigation on the effect of electrode tip on formation of metal droplets and temperature profile in a vibrating electrode electroslag remelting process
  101. Effect of North Wall Materials on the Thermal Environment in Chinese Solar Greenhouse (Part A: Experimental Researches)
  102. Three-dimensional optimal design of a cooled turbine considering the coolant-requirement change
  103. Theoretical analysis of particle size re-distribution due to Ostwald ripening in the fuel cell catalyst layer
  104. Effect of phase change materials on heat dissipation of a multiple heat source system
  105. Wetting properties and performance of modified composite collectors in a membrane-based wet electrostatic precipitator
  106. Implementation of the Semi Empirical Kinetic Soot Model Within Chemistry Tabulation Framework for Efficient Emissions Predictions in Diesel Engines
  107. Comparison and analyses of two thermal performance evaluation models for a public building
  108. A Novel Evaluation Method For Particle Deposition Measurement
  109. Effect of the two-phase hybrid mode of effervescent atomizer on the atomization characteristics
  110. Erratum
  111. Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance
  112. Erratum to: Energy converting layers for thin-film flexible photovoltaic structures
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 14.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/phys-2019-0003/html
Scroll to top button