Home On the multiplicative sum Zagreb index of molecular graphs
Article Open Access

On the multiplicative sum Zagreb index of molecular graphs

  • Xiaoling Sun EMAIL logo , Jianwei Du and Yinzhen Mei
Published/Copyright: December 17, 2024
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

Multiplicative sum Zagreb index is a modified version of the famous Zagreb indices. For a graph G , the multiplicative sum Zagreb index is defined as Π 1 * ( G ) = ∏ u v ∈ E ( G ) ( d G ( u ) + d G ( v ) ) , where E ( G ) is the edge set of G and d G ( u ) stands for the degree of vertex u in G . In this article, we determine the extremal multiplicative sum Zagreb indices among all n -vertex molecular trees, molecular unicyclic graphs, molecular bicyclic graphs and molecular tricyclic graphs.

Keywords: multiplicative sum Zagreb index; molecular tree; molecular unicyclic graphs; molecular bicyclic graph; molecular tricyclic graph
MSC 2010: 05C05; 92E10

1 Introduction

In mathematical chemistry and chemical graph theory, the topological indices are numerical parameters that can be used in describing the activities or properties of organic compounds, and they play an important role in chemistry, materials science, and pharmacology (see [1–3]). The most famous and studied topological indices are the first Zagreb index M 1 and the second Zagreb index M 2 . They first appeared within certain approximate expressions for the total π -electron energy [4]. For a graph G , the first Zagreb index and the second Zagreb index are defined as

M 1 ( G ) = ∑ u ∈ V ( G ) d G ( u ) 2 , M 2 ( G ) = ∑ u v ∈ E ( G ) d G ( u ) d G ( v ) ,

where d G ( u ) stands for the degree of vertex u in G .

These two classical topological indices ( M 1 and M 2 ) and their modified versions have been used to study heterosystems, ZE-isomerism, chirality, and complexity of molecule [5–7]. Among the modified versions, the multiplicative Zagreb indices, namely, the first and second multiplicative Zagreb indices (denoted by Π 1 and Π 2 ) [8] and multiplicative sum Zagreb index (denoted by Π 1 * ) [9] have attracted considerable attention from researchers (such as recent works [10–17]). The indices Π 1 and Π 2 are defined as follows:

Π 1 ( G ) = ∏ u ∈ V ( G ) d G ( u ) 2 , Π 2 ( G ) = ∏ u v ∈ E ( G ) d G ( u ) d G ( v ) = ∏ u ∈ V ( G ) d G ( u ) d G ( u ) ,

while the index Π 1 * is defined as

Π 1 * ( G ) = ∏ u v ∈ E ( G ) ( d G ( u ) + d G ( v ) ) .

Eliasi et al. [9] determined the minimal Π 1 * of connected graphs. Xu and Das [18] obtained the minimal and maximal Π 1 * of trees, unicyclic graphs and bicyclic graphs. Two authors of this article [11,13] presented the maximal Π 1 * with a given number of cut vertices/cut edges/vertex connectivity/edge connectivity of graphs, provided the maximal and minimal Π 1 * of trees with a fixed domination number. For other recent mathematical investigations of multiplicative sum Zagreb index, the readers can refer to [14–17].

In this article, just simple connected graphs are taken into account. For such a graph G , we represent the sets of vertices and edges by V ( G ) and E ( G ) , respectively. We denote the set of all neighbors of vertex x in G by N G ( x ) and denote the number of vertices with degree i by n i ( G ) ( n i for short). Denoted by m i , j ( G ) (or simply m i , j ), the number of edges connecting a vertex with degree i and a vertex with degree j in G . Δ ( G ) ( Δ for short) represent the maximum degree of G . Let G − u v and G + u v be the graph obtained from G by deleting the edge u v ∈ E ( G ) and by connecting the vertex u and v in G ( u v ∉ E ( G ) ). A graph G of order n is called a tree, unicyclic graph, bicyclic graph, or tricyclic graph, if it has n − 1 + r edges such that r = 0 , 1 , 2 , 3 , respectively. For terminology and notation, not defined here, we refer the readers to a relevant standard book [19].

The molecular (or chemical) graphs are graphs with d G ( x ) ≤ 4 for all x ∈ V ( G ) . In recent years, it is an important subject to study the extremal values of topological indices on molecular (or chemical) graphs [20–28]. Motivated by these works, here we provide the first 14 minimum Π 1 * of molecular trees, the first 4 minimum Π 1 * of molecular unicyclic graphs, the first 3 minimum Π 1 * of molecular bicyclic graphs and the first 7 minimum Π 1 * of molecular tricyclic graphs.

2 Some lemmas

Lemma 2.1

Let H be a graph with vertices x 1 , x 2 , x 3 , and x 4 , satisfying d H ( x 1 ) = 1 , d H ( x 2 ) = 2 , d H ( x 3 ) = 3 or 4, d H ( x 4 ) = 1 , x 1 x 2 , x 3 x 4 ∈ E ( H ) . Assume that P = y 1 y 2 … y l is a path. Denoted by G the graph obtained from H and P by attaching vertices x 1 y 1 . Let G ′ = G − x 1 y 1 + x 4 y 1 . Then, Π 1 * ( G ) > Π 1 * ( G ′ ) .

Proof

According to the definition of Π 1 * , one has

Π 1 * ( G ′ ) Π 1 * ( G ) = ( 1 + 2 ) ( d H ( x 3 ) + 2 ) ( 2 + 2 ) ( d H ( x 3 ) + 1 ) .

If d H ( x 3 ) = 3 , then Π 1 * ( G ′ ) Π 1 * ( G ) = 3 × 5 4 × 4 < 1 .

If d H ( x 3 ) = 4 , then Π 1 * ( G ′ ) Π 1 * ( G ) = 3 × 6 4 × 5 < 1 .□

Lemma 2.2

Let H 1 and H 2 be two graphs with vertices u ∈ V ( H 1 ) , v ∈ V ( H 2 ) satisfying d H 1 ( u ) = 1 or 2 and d H 2 ( v ) = 2 or 3. Assume that P 1 = x 1 x 2 … x k and P 2 = y 1 y 2 … y l are two paths. Denoted by G the graph obtained from H 1 , H 2 , P 1 , and P 2 by attaching vertices y 1 u , x 1 u , and x k v . Let G ′ = G − { x 1 u , x k v } + { u v , x 1 y l } . Then, Π 1 * ( G ) > Π 1 * ( G ′ ) .

Proof

Let us differentiate the following two cases.

Case 1. l = 1 .

Π 1 * ( G ′ ) Π 1 * ( G ) = ( 1 + 2 ) ( d H 1 ( u ) + 2 + d H 2 ( v ) + 1 ) ( d H 1 ( u ) + 3 ) ( d H 2 ( v ) + 3 ) .

If d H 1 ( u ) = 1 and d H 2 ( v ) = 2 , then Π 1 * ( G ′ ) Π 1 * ( G ) = 3 × 6 4 × 5 < 1 .

If d H 1 ( u ) = 1 and d H 2 ( v ) = 3 , then Π 1 * ( G ′ ) Π 1 * ( G ) = 3 × 7 4 × 6 < 1 .

If d H 1 ( u ) = 2 and d H 2 ( v ) = 2 , then Π 1 * ( G ′ ) Π 1 * ( G ) = 3 × 7 5 × 5 < 1 .

If d H 1 ( u ) = 2 and d H 2 ( v ) = 3 , then Π 1 * ( G ′ ) Π 1 * ( G ) = 3 × 8 5 × 6 < 1 .

Case 2. l ≥ 2 .

Π 1 * ( G ′ ) Π 1 * ( G ) = ( 2 + 2 ) ( d H 1 ( u ) + 2 + d H 2 ( v ) + 1 ) ( d H 1 ( u ) + 4 ) ( d H 2 ( v ) + 3 ) .

If d H 1 ( u ) = 1 and d H 2 ( v ) = 2 , then Π 1 * ( G ′ ) Π 1 * ( G ) = 4 × 6 5 × 5 < 1 .

If d H 1 ( u ) = 1 and d H 2 ( v ) = 3 , then Π 1 * ( G ′ ) Π 1 * ( G ) = 4 × 7 5 × 6 < 1 .

If d H 1 ( u ) = 2 and d H 2 ( v ) = 2 , then Π 1 * ( G ′ ) Π 1 * ( G ) = 4 × 7 6 × 5 < 1 .

If d H 1 ( u ) = 2 and d H 2 ( v ) = 3 , then Π 1 * ( G ′ ) Π 1 * ( G ) = 4 × 8 6 × 6 < 1 .

The proof is completed.□

Lemma 2.3

Let H be a graph with the vertex u satisfying d H ( u ) = 1 or 2. Assume that P 1 = x 1 x 2 … x k and P 2 = y 1 y 2 … y l are two paths. Denoted by G the graph obtained from H, P 1 , and P 2 by attaching vertices x 1 u and y 1 u . Let G ′ = G − x 1 u + x 1 y l . Then, Π 1 * ( G ) > Π 1 * ( G ′ ) .

Proof

Denote d H ( u ) = t = 1 or 2, N H ( u ) = { w 1 , … , w t } , d H ( w i ) = d i , 1 ≤ i ≤ t .

Case 1. k = l = 1 .

Π 1 * ( G ′ ) Π 1 * ( G ) = ( 1 + 2 ) ( t + 3 ) ( t + 3 ) ( t + 3 ) ⋅ ∏ i = 1 t d i + t + 1 d i + t + 2 < 3 ( t + 3 ) < 1

for t ≥ 1 .

Case 2. k = 1 , l ≥ 2 .

Π 1 * ( G ′ ) Π 1 * ( G ) = 2 + 2 t + 4 ⋅ ∏ i = 1 t d i + t + 1 d i + t + 2 < 4 t + 4 < 1

for t ≥ 1 .

Case 3. k ≥ 2 , l = 1 .

The proof is similar to Case 2.

Case 4. k ≥ 2 , l ≥ 2 .

Π 1 * ( G ′ ) Π 1 * ( G ) = ( 2 + 2 ) ( 2 + 2 ) ( t + 3 ) ( 1 + 2 ) ( t + 4 ) ( t + 4 ) ⋅ ∏ i = 1 t d i + t + 1 d i + t + 2 < 16 ( t + 3 ) 3 ( t + 4 ) 2 .

If t = 1 , then Π 1 * ( G ′ ) Π 1 * ( G ) < 16 × 4 3 × 25 < 1 .

If t = 2 , then Π 1 * ( G ′ ) Π 1 * ( G ) < 16 × 5 3 × 36 < 1 .

This completes the proof.□

Lemma 2.4

Let H be a molecular graph with vertices u and v satisfying d H ( u ) = 2 or 3, d H ( v ) = 2 or 3. Assume that P 1 = x 1 x 2 … x k and P 2 = y 1 y 2 … y l are two paths. Denoted by G the graph obtained from H , P 1 , and P 2 by attaching vertices x 1 u and y 1 v . Let G ′ = G − x 1 u + x 1 y l . Then, Π 1 * ( G ) > Π 1 * ( G ′ ) .

Proof

Denote d H ( u ) = t = 2 or 3, N H ( u ) = { w 1 , … , w t } , d H ( w i ) = d i , 1 ≤ i ≤ t .

Case 1. k = l = 1 .

Π 1 * ( G ′ ) Π 1 * ( G ) = ( 1 + 2 ) ( d H ( v ) + 3 ) ( d H ( v ) + 2 ) ( t + 2 ) ⋅ ∏ i = 1 t d i + t d i + t + 1 < 3 ( d H ( v ) + 3 ) ( d H ( v ) + 2 ) ( t + 2 ) .

If d H ( v ) = 2 and t = 2 , then Π 1 * ( G ′ ) Π 1 * ( G ) < 3 × 5 4 × 4 < 1 .

If d H ( v ) = 2 and t = 3 , then Π 1 * ( G ′ ) Π 1 * ( G ) < 3 × 5 4 × 5 < 1 .

If d H ( v ) = 3 and t = 2 , then Π 1 * ( G ′ ) Π 1 * ( G ) < 3 × 6 5 × 4 < 1 .

If d H ( v ) = 3 and t = 3 , then Π 1 * ( G ′ ) Π 1 * ( G ) < 3 × 6 5 × 5 < 1 .

Case 2. k = 1 , l ≥ 2 .

Π 1 * ( G ′ ) Π 1 * ( G ) = 2 + 2 t + 2 ⋅ ∏ i = 1 t d i + t d i + t + 1 < 4 t + 2 ≤ 1

for t ≥ 2 .

Case 3. k ≥ 2 , l = 1 .

Π 1 * ( G ′ ) Π 1 * ( G ) = ( 2 + 2 ) ( d H ( v ) + 3 ) ( d H ( v ) + 2 ) ( t + 3 ) ⋅ ∏ i = 1 t d i + t d i + t + 1 < 4 ( d H ( v ) + 3 ) ( d H ( v ) + 2 ) ( t + 3 ) .

If d H ( v ) = 2 and t = 2 , then Π 1 * ( G ′ ) Π 1 * ( G ) < 4 × 5 4 × 5 = 1 .

If d H ( v ) = 2 and t = 3 , then Π 1 * ( G ′ ) Π 1 * ( G ) < 4 × 5 4 × 6 < 1 .

If d H ( v ) = 3 and t = 2 , then Π 1 * ( G ′ ) Π 1 * ( G ) < 4 × 6 5 × 5 < 1 .

If d H ( v ) = 3 and t = 3 , then Π 1 * ( G ′ ) Π 1 * ( G ) < 4 × 6 5 × 6 < 1 .

Case 4. k ≥ 2 , l ≥ 2 .

Π 1 * ( G ′ ) Π 1 * ( G ) = ( 2 + 2 ) ( 2 + 2 ) ( 1 + 2 ) ( t + 3 ) ⋅ ∏ i = 1 t d i + t d i + t + 1 .

If t = 2 , then Π 1 * ( G ′ ) Π 1 * ( G ) = 16 3 × 5 ⋅ d 1 + 2 d 1 + 3 ⋅ d 2 + 2 d 2 + 3 < 16 15 ⋅ 6 7 ⋅ 6 7 < 1 .

If t = 3 , then Π 1 * ( G ′ ) Π 1 * ( G ) < 16 3 × 6 < 1 .

This completes the proof.□

3 Main results

Let M T n , M U n , M B n , and MTG n be the collections of molecular trees, molecular unicyclic graphs, molecular bicyclic graphs, and molecular tricyclic graphs of order n , respectively. Suppose that an n -vertex graph G contains r i vertices of degree d i for 1 ≤ i ≤ t , we define the degree sequence of G by D ( G ) = ( d 1 r 1 , d 2 r 2 , … , d t r t ) , where r 1 + r 2 + … + r t = n .

3.1 Molecular trees

Let Ψ ( n ) = { T ∈ ( 4 1 , 2 n − 5 , 1 4 ) ∣ m 1 , 4 ( T ) = 0 , m 2 , 2 ( T ) = n − 9 , m 1 , 2 ( T ) = m 2 , 4 ( T ) = 4 } , where n ≥ 9 , and ϒ ( n ) = { T ∈ ( 3 3 , 2 n − 8 , 1 5 ) ∣ m 1 , 3 ( T ) = 0 , m 2 , 2 ( T ) = n − 13 , m 3 , 3 ( T ) = 2 , m 1 , 2 ( T ) = m 2 , 3 ( T ) = 5 } , where n ≥ 13 .

If T ∈ Ψ ( n ) , one has

(1) Π 1 * ( T ) = 3 4 6 4 4 n − 9 ≈ 4 n − 3 ⋅ 25.6289 .

If T ∈ ϒ ( n ) , one has

(2) Π 1 * ( T ) = 3 5 5 5 6 2 4 n − 13 ≈ 4 n − 3 ⋅ 26.0711 .

Lemma 3.1

Let T ∈ M T n . Then, n 1 = n 3 + 2 n 4 + 2 and n 2 = n − 2 n 3 − 3 n 4 − 2 .

Proof

Since n 1 + n 2 + n 3 + n 4 = n and n 1 + 2 n 2 + 3 n 3 + 4 n 4 = 2 ( n − 1 ) , the result holds.□

Theorem 3.2

Consider T ′ ∈ M T n with Δ ( T ′ ) = 4 and T ′ ∉ Ψ ( n ) , where n ≥ 9 . Then, there exists T ∈ Ψ ( n ) such that Π 1 * ( T ) < Π 1 * ( T ′ ) .

Proof

We differentiate the following two cases.

Case 1. If T ′ ∈ ( 4 1 , 2 n − 5 , 1 4 ) , then T ′ satisfies at least one of the following conditions m 1 , 2 ( T ′ ) ≠ 4 , m 1 , 4 ( T ′ ) ≠ 0 , m 2 , 2 ( T ′ ) ≠ n − 9 , m 2 , 4 ( T ′ ) ≠ 4 , that is, m 1 , 2 ( T ′ ) < 4 , m 1 , 4 ( T ′ ) > 0 , m 2 , 2 ( T ′ ) > n − 9 , m 2 , 4 ( T ′ ) < 4 . By using Lemma 2.1, one can obtain a molecular tree T ∈ Ψ ( n ) such that Π 1 * ( T ) < Π 1 * ( T ′ ) .

Case 2. If T ′ ∉ ( 4 1 , 2 n − 5 , 1 4 ) , then by using Lemma 2.3, one can obtain a molecular tree T ∈ ( 4 1 , 2 n − 5 , 1 4 ) . If T ∈ Ψ ( n ) , by applying Lemma 2.3, one has Π 1 * ( T ) < Π 1 * ( T ′ ) . If T ∉ Ψ ( n ) , we can back to Case 1 and the result is true.□

Theorem 3.3

Consider T ′ ∈ M T n with Δ ( T ′ ) = 3 , n 3 ( T ′ ) ≥ 3 , and T ′ ∉ ϒ ( n ) , where n ≥ 13 . Then, there exists T ∈ ϒ ( n ) such that Π 1 * ( T ) < Π 1 * ( T ′ ) .

Proof

We differentiate the following two cases.

Case 1. If T ′ ∈ ( 3 3 , 2 n − 8 , 1 5 ) , then T ′ satisfies at least one of the following conditions m 1 , 2 ( T ′ ) ≠ 5 , m 1 , 3 ( T ′ ) ≠ 0 , m 2 , 2 ( T ′ ) ≠ n − 13 , m 2 , 3 ( T ′ ) ≠ 5 , m 3 , 3 ( T ′ ) ≠ 2 , that is, m 1 , 2 ( T ′ ) < 5 , m 1 , 3 ( T ′ ) > 0 , m 2 , 2 ( T ′ ) > n − 13 , m 2 , 3 ( T ′ ) > 5 , m 3 , 3 ( T ′ ) < 2 . By using Lemmas 2.1 and 2.2, one can obtain a molecular tree T ∈ ϒ ( n ) such that Π 1 * ( T ) < Π 1 * ( T ′ ) .

Case 2. If T ′ ∉ ( 3 3 , 2 n − 8 , 1 5 ) , since n 1 ( T ′ ) = n 3 ( T ′ ) + 2 (from Lemma 3.1) and n 3 ( T ′ ) ≥ 3 , then n 3 ( T ′ ) ≥ 4 . By using Lemma 2.3, one can obtain a molecular tree T ∈ ( 3 3 , 2 n − 8 , 1 5 ) . If T ∈ ϒ ( n ) , by applying Lemma 2.3, one has Π 1 * ( T ) < Π 1 * ( T ′ ) . If T ∉ ϒ ( n ) , we can back to Case 1 and the result holds.□

The following Theorem 3.4 determined the first 14 minimum Π 1 * of molecular trees. It is worth noting that the relevant data of Tables 1, 2, 3, 4, 5, 6, 7, 8, and 9 (also used in [20,21] recently) except the values of multiplicative sum Zagreb indices are from [22,23].

Table 1

M T n with Δ ≤ 3 , n 3 ≤ 2 and their Π 1 *

m 3 , 3 m 2 , 3 m 1 , 2 m 1 , 3 m 2 , 2 Π 1 *
A 1 0 0 2 0 n ‒ 3 4 n ‒ 3 ⋅ 9
A 2 0 1 1 2 n ‒ 5 4 n ‒ 3 ⋅ 15
A 3 0 2 2 1 n ‒ 6 4 n ‒ 3 ⋅ 14.0625
A 4 0 3 3 0 n ‒ 7 4 n ‒ 3 ⋅ 13.1836
A 5 0 2 0 4 n ‒ 7 4 n ‒ 3 ⋅ 25
A 6 0 3 1 3 n ‒ 8 4 n ‒ 3 ⋅ 23.4375
A 7 0 4 2 2 n ‒ 9 4 n ‒ 3 ⋅ 21.9727
A 8 1 1 1 3 n ‒ 7 4 n ‒ 3 ⋅ 22.5
A 9 0 5 3 1 n ‒ 10 4 n ‒ 3 ⋅ 20.5994
A 10 1 2 2 2 n ‒ 8 4 n ‒ 3 ⋅ 21.0938
A 11 0 6 4 0 n ‒ 11 4 n ‒ 3 ⋅ 19.3119
A 12 1 3 3 1 n ‒ 9 4 n ‒ 3 ⋅ 19.7754
A 13 1 4 4 0 n ‒ 10 4 n ‒ 3 ⋅ 18.5394
Table 2

Degree distributions (DD) of M U n with n 1 ≤ 2

n 4 n 3 n 2 n 1
U 1 0 0 n 0
U 2 0 1 n ‒ 2 1
U 3 1 0 n ‒ 3 2
U 4 0 2 n ‒ 4 2
Table 3

M U n with n 1 ≤ 2 and their Π 1 *

DD m 1 , 2 m 1 , 3 m 1 , 4 m 2 , 3 m 2 , 4 m 3 , 3 m 2 , 2 Π 1 *
α 1 U 1 0 0 0 0 0 0 n 4 n
α 2 U 2 0 1 0 2 0 0 n ‒ 3 4 n ⋅ 1.5625
α 3 U 2 1 0 0 3 0 0 n ‒ 4 4 n ⋅ 1.4648
α 4 U 3 0 0 2 0 2 0 n ‒ 4 4 n ⋅ 3.5156
α 5 U 3 1 0 1 0 3 0 n ‒ 5 4 n ⋅ 3.1641
α 6 U 3 2 0 0 0 4 0 n ‒ 6 4 n ⋅ 2.8477
α 7 U 4 0 2 0 2 0 1 n ‒ 5 4 n ⋅ 2.3438
α 8 U 4 1 1 0 3 0 1 n ‒ 6 4 n ⋅ 2.1973
α 9 U 4 2 0 0 4 0 1 n ‒ 7 4 n ⋅ 2.0599
α 10 U 4 0 2 0 4 0 0 n ‒ 6 4 n ⋅ 2.4414
α 11 U 4 1 1 0 5 0 0 n ‒ 7 4 n ⋅ 2.2888
α 12 U 4 2 0 0 6 0 0 n ‒ 8 4 n ⋅ 2.1458
Table 4

DD of M B n with n 1 ≤ 1

n 4 n 3 n 2 n 1
B 1 1 0 n ‒ 1 0
B 2 0 2 n ‒ 2 0
B 3 1 1 n ‒ 3 1
B 4 0 3 n ‒ 4 1
Table 5

M B n with n 1 ≤ 1 and their Π 1 *

DD m 1 , 2 m 1 , 3 m 1 , 4 m 2 , 3 m 2 , 4 m 3 , 3 m 3 , 4 m 2 , 2 Π 1 *
β 1 B 1 0 0 0 0 4 0 0 n ‒ 3 4 n ⋅ 20.2500
β 2 B 2 0 0 0 4 0 1 0 n ‒ 4 4 n ⋅ 14.6484
β 3 B 2 0 0 0 6 0 0 0 n ‒ 5 4 n ⋅ 15.2588
β 4 B 3 0 0 1 2 2 0 1 n ‒ 5 4 n ⋅ 30.7617
β 5 B 3 1 0 0 2 3 0 1 n ‒ 6 4 n ⋅ 27.6855
β 6 B 3 0 0 1 3 3 0 0 n ‒ 6 4 n ⋅ 32.9590
β 7 B 3 1 0 0 3 4 0 0 n ‒ 7 4 n ⋅ 29.6631
β 8 B 4 0 1 0 2 0 3 0 n ‒ 5 4 n ⋅ 21.0938
β 9 B 4 1 0 0 3 0 3 0 n ‒ 6 4 n ⋅ 19.7754
β 10 B 4 0 1 0 4 0 2 0 n ‒ 6 4 n ⋅ 21.9727
β 11 B 4 1 0 0 5 0 2 0 n ‒ 7 4 n ⋅ 20.5994
β 12 B 4 0 1 0 6 0 1 0 n ‒ 7 4 n ⋅ 22.8882
β 13 B 4 1 0 0 7 0 1 0 n ‒ 8 4 n ⋅ 21.4577
β 14 B 4 0 1 0 8 0 0 0 n ‒ 8 4 n ⋅ 23.8419
β 15 B 4 1 0 0 9 0 0 0 n ‒ 9 4 n ⋅ 22.3517
Table 6

DD of MTG n with n 1 ≤ 1

n 4 n 3 n 2 n 1
E 1 2 0 n ‒ 2 0
E 2 1 2 n ‒ 3 0
E 3 0 4 n ‒ 4 0
E 4 2 1 n ‒ 4 1
E 5 1 3 n ‒ 5 1
E 6 0 5 n ‒ 6 1
Table 7

MTG n with n 1 ≤ 1 and their Π 1 *

DD m 1 , 2 m 1 , 3 m 1 , 4 m 2 , 3 m 2 , 4 m 3 , 3 m 3 , 4 m 4 , 4 m 2 , 2 Π 1 *
γ 1 E 1 0 0 0 0 8 0 0 0 n ‒ 6 4 n ⋅ 410.0625
γ 2 E 1 0 0 0 0 6 0 0 1 n ‒ 5 4 n ⋅ 364.5000
γ 3 E 2 0 0 0 2 2 1 2 0 n ‒ 5 4 n ⋅ 258.3984
γ 4 E 2 0 0 0 3 3 1 1 0 n ‒ 6 4 n ⋅ 276.8555
γ 5 E 2 0 0 0 4 4 1 0 0 n ‒ 7 4 n ⋅ 296.6309
γ 6 E 2 0 0 0 4 2 0 2 0 n ‒ 6 4 n ⋅ 269.1650
γ 7 E 2 0 0 0 5 3 0 1 0 n ‒ 7 4 n ⋅ 288.3911
γ 8 E 2 0 0 0 6 4 0 0 0 n ‒ 8 4 n ⋅ 308.9905
γ 9 E 3 0 0 0 2 0 5 0 0 n ‒ 5 4 n ⋅ 189.8438
γ 10 E 3 0 0 0 4 0 4 0 0 n ‒ 6 4 n ⋅ 197.7539
γ 11 E 3 0 0 0 6 0 3 0 0 n ‒ 7 4 n ⋅ 205.9937
γ 12 E 3 0 0 0 8 0 2 0 0 n ‒ 8 4 n ⋅ 214.5767
γ 13 E 3 0 0 0 10 0 1 0 0 n ‒ 9 4 n ⋅ 223.5174
γ 14 E 3 0 0 0 12 0 0 0 0 n ‒ 10 4 n ⋅ 232.8306
γ 15 E 4 0 0 1 1 3 0 2 1 n ‒ 6 4 n ⋅ 516.7969
γ 16 E 4 1 0 0 1 4 0 2 1 n ‒ 7 4 n ⋅ 465.1172
γ 17 E 4 0 0 1 2 4 0 1 1 n ‒ 7 4 n ⋅ 553.7109
γ 18 E 4 1 0 0 2 5 0 1 1 n ‒ 8 4 n ⋅ 498.3398
γ 19 E 4 0 0 1 3 5 0 0 1 n ‒ 8 4 n ⋅ 593.2617
γ 20 E 4 1 0 0 3 6 0 0 1 n ‒ 9 4 n ⋅ 533.9355
γ 21 E 4 0 0 1 1 5 0 2 0 n ‒ 7 4 n ⋅ 581.3965
γ 22 E 4 1 0 0 1 6 0 2 0 n ‒ 8 4 n ⋅ 523.2568
γ 23 E 4 0 0 1 2 6 0 1 0 n ‒ 8 4 n ⋅ 622.9248
γ 24 E 4 1 0 0 2 7 0 1 0 n ‒ 9 4 n ⋅ 560.6323
Table 8

MTG n with n 1 ≤ 1 and their Π 1 *

DD m 1 , 2 m 1 , 3 m 1 , 4 m 2 , 3 m 2 , 4 m 3 , 3 m 3 , 4 m 4 , 4 m 2 , 2 Π 1 *
γ 25 E 4 0 0 1 3 7 0 0 0 n ‒ 9 4 n ⋅ 667.4194
γ 26 E 4 1 0 0 3 8 0 0 0 n ‒ 10 4 n ⋅ 600.6775
γ 27 E 5 0 0 1 2 0 2 3 0 n ‒ 6 4 n ⋅ 376.8311
γ 28 E 5 0 0 1 4 0 1 3 0 n ‒ 7 4 n ⋅ 392.5323
γ 29 E 5 0 0 1 6 0 0 3 0 n ‒ 8 4 n ⋅ 408.8879
γ 30 E 5 0 0 1 1 1 3 2 0 n ‒ 6 4 n ⋅ 387.5977
γ 31 E 5 0 0 1 3 1 2 2 0 n ‒ 7 4 n ⋅ 403.7476
γ 32 E 5 0 0 1 5 1 1 2 0 n ‒ 8 4 n ⋅ 420.5704
γ 33 E 5 0 0 1 7 1 0 2 0 n ‒ 9 4 n ⋅ 438.0941
γ 34 E 5 0 0 1 2 2 3 1 0 n ‒ 7 4 n ⋅ 415.2832
γ 35 E 5 0 0 1 4 2 2 1 0 n ‒ 8 4 n ⋅ 432.5867
γ 36 E 5 0 0 1 6 2 1 1 0 n ‒ 9 4 n ⋅ 450.6111
γ 37 E 5 0 0 1 8 2 0 1 0 n ‒ 10 4 n ⋅ 469.3866
γ 38 E 5 0 0 1 3 3 3 0 0 n ‒ 8 4 n ⋅ 444.9463
γ 39 E 5 0 0 1 5 3 2 0 0 n ‒ 9 4 n ⋅ 463.4857
γ 40 E 5 0 0 1 7 3 1 0 0 n ‒ 10 4 n ⋅ 482.7976
γ 41 E 5 0 0 1 9 3 0 0 0 n ‒ 11 4 n ⋅ 502.9142
γ 42 E 5 1 0 0 0 1 3 3 0 n ‒ 6 4 n ⋅ 325.5820
γ 43 E 5 1 0 0 2 1 2 3 0 n ‒ 7 4 n ⋅ 339.1479
γ 44 E 5 1 0 0 4 1 1 3 0 n ‒ 8 4 n ⋅ 353.2791
γ 45 E 5 1 0 0 6 1 0 3 0 n ‒ 9 4 n ⋅ 367.9991
γ 46 E 5 1 0 0 1 2 3 2 0 n ‒ 7 4 n ⋅ 348.8379
γ 47 E 5 1 0 0 3 2 2 2 0 n ‒ 8 4 n ⋅ 363.3728
γ 48 E 5 1 0 0 5 2 1 2 0 n ‒ 9 4 n ⋅ 378.5133
γ 49 E 5 1 0 0 7 2 0 2 0 n ‒ 10 4 n ⋅ 394.2847
γ 50 E 5 1 0 0 2 3 3 1 0 n ‒ 8 4 n ⋅ 373.7549
γ 51 E 5 1 0 0 4 3 2 1 0 n ‒ 9 4 n ⋅ 389.3280
γ 52 E 5 1 0 0 6 3 1 1 0 n ‒ 10 4 n ⋅ 405.5500
γ 53 E 5 1 0 0 8 3 0 1 0 n ‒ 11 4 n ⋅ 422.4479
γ 54 E 5 1 0 0 3 4 3 0 0 n ‒ 9 4 n ⋅ 400.4517
γ 55 E 5 1 0 0 5 4 2 0 0 n ‒ 10 4 n ⋅ 417.1371
γ 56 E 5 1 0 0 7 4 1 0 0 n ‒ 11 4 n ⋅ 434.5179
γ 57 E 5 1 0 0 9 4 0 0 0 n ‒ 12 4 n ⋅ 452.6228
Table 9

MTG n with n 1 ≤ 1 and their Π 1 *

DD m 1 , 2 m 1 , 3 m 1 , 4 m 2 , 3 m 2 , 4 m 3 , 3 m 3 , 4 m 4 , 4 m 2 , 2 Π 1 *
γ 58 E 6 0 1 0 2 0 6 0 0 n ‒ 7 4 n ⋅ 284.7656
γ 59 E 6 0 1 0 4 0 5 0 0 n ‒ 8 4 n ⋅ 296.6309
γ 60 E 6 0 1 0 6 0 4 0 0 n ‒ 9 4 n ⋅ 308.9905
γ 61 E 6 0 1 0 8 0 3 0 0 n ‒ 10 4 n ⋅ 321.8651
γ 62 E 6 0 1 0 10 0 2 0 0 n ‒ 11 4 n ⋅ 335.2761
γ 63 E 6 0 1 0 12 0 1 0 0 n ‒ 12 4 n ⋅ 349.2460
γ 64 E 6 0 1 0 14 0 0 0 0 n ‒ 13 4 n ⋅ 363.7979
γ 65 E 6 1 0 0 1 0 7 0 0 n ‒ 7 4 n ⋅ 256.2891
γ 66 E 6 1 0 0 3 0 6 0 0 n ‒ 8 4 n ⋅ 266.9678
γ 67 E 6 1 0 0 5 0 5 0 0 n ‒ 9 4 n ⋅ 278.0914
γ 68 E 6 1 0 0 7 0 4 0 0 n ‒ 10 4 n ⋅ 289.6786
γ 69 E 6 1 0 0 9 0 3 0 0 n ‒ 11 4 n ⋅ 301.7485
γ 70 E 6 1 0 0 11 0 2 0 0 n ‒ 12 4 n ⋅ 314.3214
γ 71 E 6 1 0 0 13 0 1 0 0 n ‒ 13 4 n ⋅ 327.4181
γ 72 E 6 1 0 0 15 0 0 0 0 n ‒ 14 4 n ⋅ 341.0605

Theorem 3.4

For n ≥ 13 , T 1 ∈ A 1 , T 2 ∈ A 4 , T 3 ∈ A 3 , T 4 ∈ A 2 , T 5 ∈ A 13 , T 6 ∈ A 11 , T 7 ∈ A 12 , T 8 ∈ A 9 , T 9 ∈ A 10 , T 10 ∈ A 7 , T 11 ∈ A 8 , T 12 ∈ A 6 , T 13 ∈ A 5 , T 14 ∈ Ψ ( n ) , T 15 ∈ ϒ ( n ) , and T ∈ M T n \ { T 1 , T 2 , … , T 14 } , then Π 1 * ( T 1 ) < Π 1 * ( T 2 ) < Π 1 * ( T 3 ) < … < Π 1 * ( T 14 ) < Π 1 * ( T ) .

Proof

By using Table 1 and the Π 1 * of molecular trees among Ψ ( n ) and ϒ ( n ) , we have Π 1 * ( T 1 ) < Π 1 * ( T 2 ) < Π 1 * ( T 3 ) < … < Π 1 * ( T 14 ) < Π 1 * ( T 15 ) . If Δ ( T ) = 4 , by Theorem 3.2, we have Π 1 * ( T 14 ) < Π 1 * ( T ) . If Δ ( T ) ≤ 3 and n 3 ( T ) ≤ 2 , by Table 1, one has Π 1 * ( T 1 ) < Π 1 * ( T 2 ) < Π 1 * ( T 3 ) < … < Π 1 * ( T 13 ) < Π 1 * ( T ) . If Δ ( T ) = 3 and n 3 ( T ) ≥ 3 , then by Theorem 3.3, we have Π 1 * ( T 15 ) < Π 1 * ( T ) . Combining above arguments and equations (1) and (2), the theorem holds.□

3.2 Molecular unicyclic graphs

Lemma 3.5

[22,23] U belongs to one of the equivalence classes given in Table 2 if and only if U ∈ M U n with n 1 ( U ) ≤ 2 .

Theorem 3.6

For n ≥ 7 , G 1 ∈ α 1 , G 2 ∈ α 3 , G 3 ∈ α 2 , G 4 ∈ α 9 in Table 3, and U ∈ M U n \ { G 1 , G 2 , G 3 , G 4 } , then Π 1 * ( G 1 ) < Π 1 * ( G 2 ) < Π 1 * ( G 3 ) < Π 1 * ( G 4 ) < Π 1 * ( U ) .

Proof

By using Table 3, one has Π 1 * ( G 1 ) < Π 1 * ( G 2 ) < Π 1 * ( G 3 ) < Π 1 * ( G 4 ) . If n 1 ( U ) ≤ 2 , by using Tables 2 and 3, one can obtain the desired result. If n 1 ( U ) ≥ 3 , then by Lemmas 2.3 and 2.4, one can obtain a molecular unicyclic graph U ′ with n 1 ( U ′ ) = 2 such that Π 1 * ( U ) > Π 1 * ( U ′ ) . Using Table 3, Π 1 * ( G 4 ) ≤ Π 1 * ( U ′ ) . Hence, the result holds.□

3.3 Molecular bicyclic graphs

Lemma 3.7

[22,23] B belongs to one of the equivalence classes given in Table 4  if and only if B ∈ M B n with n 1 ( B ) ≤ 1 .

Theorem 3.8

For n ≥ 6 , G 1 ∈ β 2 , G 2 ∈ β 3 , G 3 ∈ β 9 in Table 5, and B ∈ M B n \ { G 1 , G 2 , G 3 } , then Π 1 * ( G 1 ) < Π 1 * ( G 2 ) < Π 1 * ( G 3 ) < Π 1 * ( B ) .

Proof

By using Table 5, one has Π 1 * ( G 1 ) < Π 1 * ( G 2 ) < Π 1 * ( G 3 ) . If n 1 ( B ) ≤ 1 , by using Table 5, one can obtain the desired result. If n 1 ( B ) ≥ 2 , then by Lemmas 2.3 and 2.4, one can obtain a molecular bicyclic graph B ′ with n 1 ( B ′ ) = 1 such that Π 1 * ( B ) > Π 1 * ( B ′ ) . Using Table 5, Π 1 * ( G 3 ) ≤ Π 1 * ( B ′ ) . Hence, the result holds.□

3.4 Molecular tricyclic graphs

Lemma 3.9

[22,23] TG belongs to one of the equivalence classes given in Table 6 if and only if T G ∈ MTG n with n 1 ( T G ) ≤ 1 .

Theorem 3.10

For n ≥ 6 , G 1 ∈ γ 9 , G 2 ∈ γ 10 , G 3 ∈ γ 11 , G 4 ∈ γ 12 , G 5 ∈ γ 13 , G 6 ∈ γ 14 , G 7 ∈ γ 65 in Tables 7, 8, 9, and T G ∈ MTG n \ { G 1 , G 2 , G 3 , G 4 , G 5 , G 6 , G 7 } , then Π 1 * ( G 1 ) < Π 1 * ( G 2 ) < Π 1 * ( G 3 ) < Π 1 * ( G 4 ) < Π 1 * ( G 5 ) < Π 1 * ( G 6 ) < Π 1 * ( G 7 ) < Π 1 * ( T G ) .

Proof

By using Tables 7–9, one has Π 1 * ( G 1 ) < Π 1 * ( G 2 ) < Π 1 * ( G 3 ) < Π 1 * ( G 4 ) < Π 1 * ( G 5 ) < Π 1 * ( G 6 ) < Π 1 * ( G 7 ) . If n 1 ( T G ) ≤ 1 , by using Tables 7–9, one can obtain the desired result. If n 1 ( T G ) ≥ 2 , then by Lemmas 2.3 and 2.4, one can obtain a molecular tricyclic graph T G ′ with n 1 ( T G ′ ) = 1 such that Π 1 * ( T G ) > Π 1 * ( T G ′ ) . Using Tables 7–9, Π 1 * ( G 7 ) ≤ Π 1 * ( T G ′ ) . Hence, the result holds.□

Acknowledgements

The authors would like to thank the anonymous editors and reviewers for helpful comments and suggestions that improved the original version of the article.

  1. Funding information: This work was supported by the Natural Science Foundation of Shanxi Province of China (No. 202303021211154) and the Shanxi Scholarship Council of China (No. 2022-149).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. XS provided the funding support, supervised and led the planning and execution of this research, and reviewed and evaluated the manuscript. JD proposed the research idea, formed the overall research objective, and wrote the first draft. YM provided the funding support, revised the manuscript and made some useful investigations. XS and JD revised the final draft.

  3. Conflict of interest: The authors state no conflicts of interest.

  4. Data availability statement: The article does not include or use any datasets.

References

[1] I. Gutman and B. Furtula, Novel Molecular Structure Descriptors - Theory and Applications I, Univ. Kragujevac, Kragujevac, 2010. Search in Google Scholar

[2] I. Gutman and B. Furtula, Novel Molecular Structure Descriptors - Theory and Applications II, Univ. Kragujevac, Kragujevac, 2010. Search in Google Scholar

[3] R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2000. 10.1002/9783527613106Search in Google Scholar

[4] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. III. Total π electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538, DOI: https://doi.org/10.1016/0009-2614(72)85099-1. 10.1016/0009-2614(72)85099-1Search in Google Scholar

[5] J. Braun, A. Kerber, M. Meringer, and C. Rucker, Similarity of molecular descriptors: The equivalence of Zagreb indices and walk counts, MATCH Commun. Math. Comput. Chem. 54 (2005), no. 1, 163–176. Search in Google Scholar

[6] S. Nikolić, G. Kovaćević, A. Milicević, and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003), no. 2, 113–124. Search in Google Scholar

[7] S. Nikolić, I. M. Tolić, N. Trinajstić, and I. Baućic, On the Zagreb indices as complexity indices, Croat. Chem. Acta 73 (2000), no. 4, 909–921. Search in Google Scholar

[8] R. Todeschini and V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem. 64 (2010), no. 2, 359–372. Search in Google Scholar

[9] M. Eliasi, A. Iranmanesh, and I. Gutman, Multiplicative versions of first Zagreb index, MATCH Commun. Math. Comput. Chem. 68 (2012), no. 1, 217–230. Search in Google Scholar

[10] J. Du and X. Sun, Extremal quasi-unicyclic graphs with respect to the general multiplicative Zagreb indices, Discrete Appl. Math. 325 (2023), 200–211, DOI: https://doi.org/10.1016/j.dam.2022.10.019. 10.1016/j.dam.2022.10.019Search in Google Scholar

[11] J. Du and X. Sun, On the multiplicative sum Zagreb index of graphs with some given parameters, J. Math. Inequal. 14 (2020), no. 4, 1165–1181, DOI: https://doi.org/10.7153/jmi-2020-14-76. 10.7153/jmi-2020-14-76Search in Google Scholar

[12] J. Du and X. Sun, Quasi-tree graphs with extremal general multiplicative Zagreb indices, IEEE Access 8 (2020), 194676–194684, DOI: https://doi.org/10.1109/access.2020.3033929. 10.1109/ACCESS.2020.3033929Search in Google Scholar

[13] X. Sun, Y. Gao, and J. Du, On multiplicative sum Zagreb index of trees with fixed domination number, J. Math. Inequal. 17 (2023), no. 1, 83–98, DOI: https://doi.org/10.7153/jmi-2023-17-06. 10.7153/jmi-2023-17-06Search in Google Scholar

[14] C. Xu, B. Horoldagva, and L. Buyantogtokh, Cactus graphs with maximal multiplicative sum Zagreb index, Symmetry 13 (2021), no. 5, 913, DOI: https://doi.org/10.3390/sym13050913. 10.3390/sym13050913Search in Google Scholar

[15] M. Azari and A. Iranmanesh, Some inequalities for the multiplicative sum Zagreb index of graph operations, J. Math. Inequal. 9 (2015), no. 3, 727–738, DOI: https://doi.org/10.7153/jmi-09-60. 10.7153/jmi-09-60Search in Google Scholar

[16] V. Božović, Ž. K. Kovijanić, and G. Popivoda, Chemical trees with extreme values of a few types of multiplicative Zagreb indices, MATCH Commun. Math. Comput. Chem. 76 (2016), no. 1, 207–220. Search in Google Scholar

[17] B. Horoldagva, C. Xu, L. Buyantogtokh, and S. Dorjsembe, Extremal graphs with respect to the multiplicative sum Zagreb index, MATCH Commun. Math. Comput. Chem. 84 (2020), no. 3, 773–786. Search in Google Scholar

[18] K. Xu and K. C. Das, Trees, unicyclic, and bicyclic graphs extremal with respect to multiplicative sum Zagreb index, MATCH Commun. Math. Comput. Chem. 68 (2012), no. 1, 257–272. Search in Google Scholar

[19] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Elsevier, New York, 1976. 10.1007/978-1-349-03521-2Search in Google Scholar

[20] X. Zuo, A. Jahanbani, and H. Shooshtari, On the atom-bond sum-connectivity index of chemical graphs, J. Mol. Struct. 1296 (2024), no. 1, 136849, DOI: https://doi.org/10.1016/j.molstruc.2023.136849. 10.1016/j.molstruc.2023.136849Search in Google Scholar

[21] H. Liu, L. You, and Y. Huang, Ordering chemical graphs by Sombor indices and its applications, MATCH Commun. Math. Comput. Chem. 87 (2022), no. 1, 5–22, DOI: https://doi.org/10.46793/match.87-1.005l. 10.46793/match.87-1.005LSearch in Google Scholar

[22] A. Ghalavand and A. R. Ashrafi, Ordering chemical graphs by Randić and sum-connectivity numbers, Appl. Math. Comput. 331 (2018), 160–168, DOI: https://doi.org/10.1016/j.amc.2018.02.049. 10.1016/j.amc.2018.02.049Search in Google Scholar

[23] I. Gutman, A. Ghalavand, T. Dehghan-Zadeh, and A. R. Ashrafi, Graphs with smallest forgotten index, Iranian J. Math. Chem. 8 (2017), no. 3, 259–273. Search in Google Scholar

[24] A. R. Ashrafi and A. Ghalavand, Ordering chemical trees by Wiener polarity index, Appl. Math. Comput. 313 (2017), 301–312, DOI: https://doi.org/10.1016/j.amc.2017.06.005. 10.1016/j.amc.2017.06.005Search in Google Scholar

[25] A. Ghalavand and A. R. Ashrafi, Ordering of c-cyclic graphs with respect to total irregularity, J. Appl. Math. Comput. 63 (2020), 707–715, DOI: https://doi.org/10.1007/s12190-020-01335-6. 10.1007/s12190-020-01335-6Search in Google Scholar

[26] A. Ali, Z Du, and M. Ali, A note on chemical trees with minimum Wiener polarity index, Appl. Math. Comput. 335 (2018), 231–236, DOI: https://doi.org/10.1016/j.amc.2018.04.051. 10.1016/j.amc.2018.04.051Search in Google Scholar

[27] Y. Jiang, X. Chen, and W. Lin, A note on chemical trees with maximal inverse sum indeg index, MATCH Commun. Math. Comput. Chem. 86 (2021), no. 1, 29–38. Search in Google Scholar

[28] A. Ghalavand and A. R. Ashrafi, Extremal graphs with respect to variable sum exdeg index via majorization, Appl. Math. Comput. 303 (2017), 19–23, DOI: https://doi.org/10.1016/j.amc.2017.01.007. 10.1016/j.amc.2017.01.007Search in Google Scholar
Received: 2024-03-29
Revised: 2024-11-23
Accepted: 2024-11-24
Published Online: 2024-12-17

© 2024 the author(s), published by De Gruyter

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

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2024-0108
Keywords for this article
multiplicative sum Zagreb index; molecular tree; molecular unicyclic graphs; molecular bicyclic graph; molecular tricyclic graph
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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/math-2024-0108/html
Scroll to top button