General Randić indices of a graph and its line graph

Yan Liang; Baoyindureng Wu

doi:10.1515/math-2022-0611

Article Open Access

General Randić indices of a graph and its line graph

Yan Liang and Baoyindureng Wu

Published/Copyright: August 1, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

For a real number α , the general Randić index of a graph G , denoted by R α ( G ) , is defined as the sum of ( d ( u ) d ( v ) ) α for all edges u v of G , where d ( u ) denotes the degree of a vertex u in G . In particular, R − 1 2 ( G ) is the ordinary Randić index, and is simply denoted by R ( G ) . Let α be a real number. In this article, we show that

if α ≥ 0 , R α ( L ( G ) ) ≥ 2 R α ( G ) for any graph G with δ ( G ) ≥ 3 ;
if α ≥ 0 , R α ( L ( G ) ) ≥ R α ( G ) for any connected graph G which is not isomorphic to P n ;
if α < 0 , R α ( L ( G ) ) ≥ R α ( G ) for any k -regular graph G with k ≥ 2 − 2 α + 1 ;
R ( L ( S ( G ) ) ) ≥ R ( S ( G ) ) for any graph G with δ ( G ) ≥ 3 , where S ( G ) is the graph obtained from G by inserting exactly one vertex into each edge.

Keywords: general Randić index; line graph; Randić index

MSC 2010: 05C09; 05C76

1 Introduction

Throughout this work, let G = ( V ( G ) , E ( G ) ) be a graph of order n and of size m , where n = ∣ V ( G ) ∣ and m = ∣ E ( G ) ∣ . For a vertex v ∈ V ( G ) , d G ( v ) denotes the degree of v in G . A graph G is called k-regular if any v ∈ V ( G ) has degree k , where k is a positive integer. As usual, Δ ( G ) and δ ( G ) represent the maximum and the minimum degree of G , respectively. That is, Δ ( G ) = max { d G ( v ) : v ∈ V ( G ) } and δ ( G ) = min { d G ( v ) : v ∈ V ( G ) } . Furthermore, N G ( u ) denotes the neighborhood of a vertex u in G , and N G [ u ] denotes the closed neighborhood of a vertex u in G , i.e., N G [ u ] = N G ( u ) ∪ { u } . For v , u ∈ V ( G ) , we denote by d G ( u , v ) (or simply d ( u , v ) ), the length of a shortest path in G between u and v . For any u ∈ V ( G ) , I ( u ) denotes the set of edges incident to u in G .

A topological descriptor is a single number that represents a chemical structure in graph-theoretical terms via the molecular graph. It is called a topological index if besides this descriptor correlates with a molecular property, and then it can be used to understand physicochemical properties of chemical compounds. In 1947, Wiener [1] introduced a parameter, denoted by W ( G ) , of a connected graph G , now widely known as the Wiener index, since then hundreds of topological indices have been introduced and studied. Precisely,

W ( G ) = ∑ { u , v } ⊆ V ( G ) d G ( u , v ) .

Das and Nadjafi-Arani [2] and Chen et al. [3] studied the bounds for Wiener indices of graphs with given order and radius. In [4–7], Wiener indices of digraphs were investigated. We refer to [8] for the mathematical aspect of Wiener index of graphs. Spiro [9] recently introduced the Wiener index for a signed graph and conjectured that the path P n with alternating signs has the minimum Wiener index among all signed trees with vertices. By constructing an infinite family of counterexamples, Guo et al. [10] proved that the conjecture is false whenever n is at least 30. A relation between Wiener index of a graph and that of its line graph was investigated in [11–15].

In 1975, Randić [16] introduced a graph parameter, now called the Randić index R ( G ) . For a graph G ,

R ( G ) = ∑ u v ∈ E ( G ) 1 d ( u ) d ( v ) .

As a generalization of the Randić index, the general Randić index of a graph G was introduced by Gutman and Pavlović [13] in 1997. That is,

R α ( G ) = ∑ u v ∈ E ( G ) ( d ( u ) d ( v ) ) α ,

where α is a real number. The relations between R α and the general zeroth-order Randić index Q α were investigated in [17,18], where

Q α = Q α ( G ) = ∑ u ∈ V ( G ) d ( u ) ≥ 1 d ( u ) α .

In fact, there exist numerous papers and a series of books or surveys about this molecular descriptor [19–25] and the references therein. Carbollosa et al. [26] introduced the f-index and the f-polynomial of a graph and obtained inequalities involving the f-polynomial of many graph operations including the corona product graph, the join graph, and line graph and the Mycielskian graph. On the basis of the work of Bollobás and Erdös [27] and Bollobás et al. [28] gave an upper bound for R α when α > 0 and a lower bound for R α when α < 0 in terms of the size of G . Lu et al. [29] gave an upper bound for R α when 0 < α ≤ 1 and a lower bound for R α when − 1 ≤ α < 0 using the order, the size, and the maximum eigenvalue of the adjacency matrix of G . Li and Yang [30] obtained some bounds for R α of a graph using the order of G . Hu et al. [31] studied R α of a tree. Clark and Moon [32] found the expected value and variance of R α for certain families of trees. Rodríguez and Sigarreta [24] derived relations between R α 1 and R α 2 for different values of α 1 and α 2 . Liu and Gutman [17] gave some lower bounds for R α when α > 1 and upper bounds for R α when α < − 1 . Zhang and Wu [33] showed that for any tree T of order n ≥ 3 , R ( L ( T ) ) > n 4 . In this article, we show a relationship of the general Randić index between a graph and that of its line graph for various value of α .

The second Zagreb indices is defined as follows:

M 2 ( G ) = ∑ u v ∈ E ( G ) d ( u ) d ( v ) .

We refer to [34–36] for some relevant work on it. Clearly, M 2 ( G ) = R 1 ( G ) for a graph G . Another remarkable topological descriptor is the harmonic index, defined in [37] as follows:

H ( G ) = ∑ u v ∈ E ( G ) 2 d ( u ) + d ( v ) .

An important recent survey with a very complete chapter on the harmonic index is [38]. Its chemical applicability has begun to be investigated, for example, in [20,39] and regarding physical and chemical properties, it has been discovered that H ( G ) is similar in correlation to the well-known Randić index.

Line graphs were initially introduced in [40,41], but the terminology was used in [42] for the first time. The line graph of a graph G , denoted by L ( G ) , is the graph with V ( L ( G ) ) = E ( G ) , in which two vertices are adjacent if and only if they share a common end vertex in G . Line graphs play an important role in the study of topological indices [43–48]. Line graphs can be used in solving maximization and minimization problems on hexagonal systems since it can be transformed into one on triangular systems [49].

As usual, we denote the complete graph, the path, the star, and the cycle of order n by K n , P n , K 1 , n − 1 , and C n , respectively. A vertex of degree one is called a l e a f , and a vertex adjacent to a leaf is called a s t e m .

Theorem 1.1

Let G be a connected graph of order n and size m.

(Buckley [11]) If G is a tree, then W ( L ( T ) ) = W ( T ) − n 2 ;
(Gutman and Pavlović [13]) If G is an unicyclic graph, then W ( L ( G ) ) ≤ W ( G ) with equality if and only if G ≅ C n ;
(Cohen et al. [12] and Wu [15]) If G is a connected graph with δ ( G ) ≥ 2 , then W ( G ) ≤ W ( L ( G ) ) , with equality if and only if G ≅ C n .

Inspired from Theorem 1.1 and Nadeem et al. [50], we establish the relation between the general Randić indices of graphs and their line graphs. In addition, we consider the relation between the Randić indices of subdivision graphs and their line graphs.

Theorem 1.2

Let α ≥ 0 be a real number and G be a connected graph of order n . If G ≇ P n , then R α ( L ( G ) ) ≥ R α ( G ) . Furthermore, if δ ( G ) ≥ 3 , then R α ( L ( G ) ) ≥ 2 R α ( G ) .

The reason for the assumption that G is not isomorphic to P n is that L ( P n ) = P n − 1 and then R α ( L ( P n ) ) ≤ R α ( P n ) when α ≥ 0 .

Theorem 1.3

Let k be a positive integer and G be a k-regular graph of order n. For any given real number α < 0 , if k ≥ 2 − 2 α + 1 , then R α ( L ( G ) ) ≥ R α ( G ) .

In particular, a relationship between R ( G ) and R ( L ( G ) ) is given in the next theorem.

Theorem 1.4

Let G be a connected graph with δ ( G ) ≥ 2 . If Δ ( G ) ≤ δ ( G ) 2 − 2 δ ( G ) + 2 , then R ( L ( G ) ) ≥ R ( G ) .

The subdivision graph S ( G ) of G is obtained by inserting a new vertex of degree two on each edge of G and S 2 ( G ) of G is obtained by subdividing each edge of G at least once. If G has n vertices and m edges, then S ( G ) has n + m vertices and 2 m edges. Clearly, S ( G ) is bipartite.

Theorem 1.5

For any graph G, the following statements hold:

If δ ( G ) ≥ 3 , then R ( L ( S ( G ) ) ) > R ( S ( G ) ) .
If δ ( G ) ≥ 6 , then R ( L ( S 2 ( G ) ) ) > R ( S 2 ( G ) ) .

In the next section, we present several lemmas, and in Section 3, we prove these aforementioned theorems using the lemmas proved in Section 2.

2 Preliminary results

Let us present several useful lemmas.

Lemma 2.1

Let α ≥ 0 be a real number. For any integer n ≥ 4 ,

R α ( L ( K 1 , n − 1 ) ) ≥ R α ( K 1 , n − 1 ) ,

with equality if and only if α = 0 and n = 4 .

Proof

Since L ( K 1 , n − 1 ) ≅ K n − 1 , we have

R α ( L ( K 1 , n − 1 ) ) = R α ( K n − 1 ) = n − 1 2 ( n − 2 ) 2 α ,

and

R α ( K 1 , n − 1 ) = ( n − 1 ) α + 1 .

Then

R α ( K n − 1 ) R α ( K 1 , n − 1 ) = ( n − 2 ) 2 α + 1 2 ( n − 1 ) α .

It suffices to show that ( n − 2 ) 2 α + 1 ≥ 2 ( n − 1 ) α , equivalently ( n − 2 ) α + 1 − 2 1 + 1 n − 2 α ≥ 0 .

Thus, it is natural to consider the function f ( x ) = x α + 1 − 2 1 + 1 x α . Since α ≥ 0 , f ′ ( x ) = ( α + 1 ) x α + 2 α 1 + 1 x α − 1 ⋅ 1 x 2 ≥ 0 for any x ≥ 2 , implying that f ( x ) increasing on the interval [ 2 , + ∞ ) . Since n ≥ 4 , f ( n − 2 ) ≥ f ( 2 ) . Moreover, since f ( 2 ) = 2 α + 1 − 2 ⋅ 3 2 α ≥ 0 , we have

( n − 2 ) α + 1 − 2 1 + 1 n − 2 α = f ( n − 2 ) ≥ 0 ,

with equality if and only if α = 0 and n = 4 , as we desired.□

For simplicity, d L ( G ) ( e ) is denoted by d L ( e ) for any e ∈ E ( G ) .

Lemma 2.2

Let α ≥ 0 . For a vertex x of a connected graph G of order n with d ( x ) ≥ 3 ,

∑ y ∈ N ( x ) ( d ( x ) d ( y ) ) α ≤ ∑ e , f ∈ I ( x ) e ≠ f ( d L ( e ) d L ( f ) ) α ,

with equality if and only if α = 0 , G ≅ K 1 , 3 and x is the center of K 1 , 3 .

Proof

Let N ( x ) = { x 1 , x 2 , … , x t } where t = d ( x ) . For convenience, e j = x x j for j satisfying 1 ≤ j ≤ t . Assume that d ( x 1 ) ≤ d ( x 2 ) ≤ … ≤ d ( x t ) for 1 ≤ j ≤ t .

If d ( x t ) = 1 , then by the assumption that G is connected, we have G ≅ K 1 , t and t = n − 1 . By Lemma 2.1, the result holds immediately. Now let d ( x t ) ≥ 2 . By using our notation,

∑ y ∈ N ( x ) ( d ( x ) d ( y ) ) α = ∑ i = 1 t ( d ( x ) d ( x i ) ) α ,

and

∑ e , f ∈ I ( x ) e ≠ f ( d L ( e ) d L ( f ) ) α = ∑ 1 ≤ i < j ≤ t ( d L ( e i ) d L ( e j ) ) α = ∑ 1 ≤ i < j ≤ t ( d ( x ) + d ( x i ) − 2 ) α ( d ( x ) + d ( x j ) − 2 ) α ≥ ∑ 1 ≤ i < j ≤ t ( d ( x i ) d ( x ) ) α ,

with equality if and only if d ( x t ) = 1 , t = 3 , and α = 0 .

Since t ≥ 3 , t 2 ≥ t , it follows that

∑ 1 ≤ i < j ≤ t ( d ( x i ) d ( x ) ) α ≥ ∑ i = 1 t ( d ( x ) d ( x i ) ) α ,

with equality if and only if d ( x t ) = 1 , t = 3 , i.e., G ≅ K 1 , 3 , and α = 0 .□

Let G be a connected graph not isomorphic to a path. A path P = x 0 x 1 ⋯ x k of G with k ≥ 2 is said to be a 2-extremal path if d ( x 0 ) ≠ 2 , d ( x k ) ≠ 2 and d ( x 1 ) = ⋯ = d ( x k − 1 ) = 2 . We use P to denote the set of 2-extremal paths in G . For any path P ∈ P , let P ≥ 3 be the set of ends with degree at least 3 in G . Clearly, 1 ≤ ∣ P ≥ 3 ∣ ≤ 2 .

Lemma 2.3

Let α ≥ 0 be a real number. If P ∈ P of a connected graph G not isomorphic to a path, then

(1) ∑ x y ∈ E ( P \ P ≥ 3 ) ( d ( x ) d ( y ) ) α ≤ ∑ e f ∈ E ( L ( G ) ) e , f ∈ E ( P ) , e ≠ f ( d L ( e ) d L ( f ) ) α .

Proof

Since G is not a path, P can be labeled as x 0 x 1 ⋯ x k , where d ( x 0 ) ≥ 3 . Let e i = x i − 1 x i for each i ∈ { 1 , … , k } . Thus,

∑ e f ∈ E ( L ( G ) ) e , f ∈ E ( P ) , e ≠ f ( d L ( e ) d L ( f ) ) α = ∑ i = 1 k − 1 ( d L ( e i ) d L ( e i + 1 ) ) α ,

and

∑ x y ∈ E ( P \ P ≥ 3 ) ( d ( x ) d ( y ) ) α = ∑ i = 1 k − 2 ( d ( x i ) d ( x i + 1 ) ) α + g ( x k ) ⋅ 2 α ,

where

g ( x k ) = 1 , if d ( x k ) = 1 , 0 , if d ( x k ) ≥ 3 .

Since d ( x 1 ) = d ( x 2 ) = ⋯ = d ( x k − 1 ) = 2 , we have

(2) ∑ i = 1 k − 2 ( d ( x i ) d ( x i + 1 ) ) α + g ( x k ) ⋅ 2 α = ( k − 2 ) ⋅ 4 α + g ( x k ) ⋅ 2 α ,

and

(3) ∑ i = 1 k − 1 ( d L ( e i ) d L ( e i + 1 ) ) α = 2 α ⋅ d ( x 0 ) α + 2 α ⋅ d ( x k ) α + ( k − 3 ) ⋅ 4 α , if k ≥ 3 , ( d ( x 0 ) d ( x k ) ) α , if k = 2 .

Then

( 3 ) − ( 2 ) = 2 α ⋅ d ( x 0 ) α − 4 α + ( d ( x k ) α − g ( x k ) ) ⋅ 2 α , if k ≥ 3 , d ( x 0 ) α − g ( x k ) ⋅ 2 α , if k = 2 .

Since d ( x 0 ) ≥ 3 and α ≥ 0 , by the aforementioned expression,

∑ i = 1 k − 1 ( d L ( e i ) d L ( e i + 1 ) ) α − ∑ i = 1 k − 2 ( d ( x i ) d ( x i + 1 ) ) α − g ( x k ) ⋅ 2 α ≥ 0 .

The proof is now finished.□

3 Proofs

Proof of Theorem 1.2

Observe that

R α ( G ) = ∑ x y ∈ E ( G ) ( d ( x ) d ( y ) ) α = 1 2 ∑ x ∈ V ( G ) ∑ y ∈ N ( x ) ( d ( x ) d ( y ) ) α

and

R α ( L ( G ) ) = ∑ x ∈ V ( G ) ∑ e , f ∈ I ( x ) e ≠ f ( d L ( e ) d L ( f ) ) α .

For each x ∈ V ( G ) with d ( x ) ≥ 3 , by Lemma 2.2,

∑ y ∈ N ( x ) ( d ( x ) d ( y ) ) α ≤ ∑ e , f ∈ I ( x ) e ≠ f ( d L ( e ) d L ( f ) ) α .

Summing up the aforementioned facts, we conclude that R α ( L ( G ) ) ≥ 2 R α ( G ) if δ ( G ) ≥ 3 .

We consider the case when δ ( G ) ≤ 2 . If G ≅ C n , then L ( G ) ≅ C n , and thus, R α ( L ( G ) ) = R α ( G ) . Note that

R α ( G ) ≤ ∑ x ∈ V ( G ) d ( x ) ≥ 3 ∑ y ∈ N ( x ) ( d ( x ) d ( y ) ) α + ∑ P ∈ P ∑ x y ∈ E ( P \ P ≥ 3 ) ( d ( x ) d ( y ) ) α

and

R α ( L ( G ) ) = ∑ x ∈ V ( G ) d ( x ) ≥ 3 ∑ e , f ∈ I ( x ) e ≠ f ( d L ( e ) d L ( f ) ) α + ∑ P ∈ P ∑ e f ∈ E ( L ( G ) ) e , f ∈ E ( P ) , e ≠ f ( d L ( e ) d L ( f ) ) α .

For each x ∈ V ( G ) with d ( x ) ≥ 3 , by Lemma 2.2,

∑ y ∈ N ( x ) ( d ( x ) d ( y ) ) α ≤ ∑ e , f ∈ I ( x ) e ≠ f ( d L ( e ) d L ( f ) ) α .

For P ∈ P , by Lemma 2.3,

∑ x y ∈ E ( P \ P ≥ 3 ) ( d ( x ) d ( y ) ) α ≤ ∑ e f ∈ E ( L ( G ) ) e , f ∈ E ( P ) , e ≠ f ( d L ( e ) d L ( f ) ) α .

Combining the aforementioned equations, we conclude that R α ( L ( G ) ) ≥ R α ( G ) if δ ( G ) ≤ 2 .□

Proof of Theorem 1.3

Because G is a k -regular graph of order n , the line graph L ( G ) of G is a ( 2 k − 2 ) -regular graph of order k n 2 . By the definition of the general Randić index of a graph G , it follows that

R α ( L ( G ) ) = 1 2 ⋅ n k ( 2 k − 2 ) 2 ⋅ ( 2 k − 2 ) 2 α , R α ( G ) = n k 2 α + 1 2 .

Therefore,

R α ( L ( G ) ) R α ( G ) = ( k − 1 ) 2 − 2 k 2 α .

Let α < 0 and k ≥ 2 . Then the function f ( x ) = x 2 α is monotonously decreasing and ( k − 1 ) ( 2 − 2 k ) 2 α > ( k − 1 ) ⋅ 2 2 α . Therefore, R α ( L ( G ) ) R α ( G ) > 1 for k ≥ 2 − 2 α + 1 , as we desired.□

Proof of Theorem 1.4

Let V ( G ) = { v 1 , v 2 , … , v n } , t i = d G ( v i ) , N ( v i ) = { v i 1 , v i 2 , … , v i t i } and d i j = d G ( v i j ) . By the definition of line graph and Randić index, we have

(4) R ( L ( G ) ) − R ( G ) = ∑ i = 1 n ∑ 1 ≤ j 1 < j 2 ≤ t i 1 t i + d i j 1 − 2 t i + d i j 2 − 2 − ∑ i = 1 t i 1 2 t i d i j ≥ ∑ i = 1 n t i ( t i − 1 ) 2 ( t i + Δ ( G ) − 2 ) − t i 2 t i δ ( G ) = ∑ i = 1 n t i 2 t i − 1 ( t i + Δ ( G ) − 2 ) − 1 t i δ ( G ) .

Since R ( L ( G ) ) − R ( G ) ≥ 0 , we want to identify (4) is positive. It follows that

Δ ( G ) ≤ ( t − 1 ) t δ ( G ) − t + 2 .

Set f ( t ) = ( t − 1 ) t δ ( G ) − t + 2 . We know that the function f ( t ) is an increasing function on interval [ 1 , + ∞ ) , since

f ′ ( x ) = t δ ( G ) + δ ( G ) 2 t − 1 t − 1 ≥ 0 .

Thus, we have that Δ ( G ) ≤ f ( δ ( G ) ) = δ ( G ) 2 − 2 δ ( G ) + 2 . This proof is complete.□

Proof of Theorem 1.5

Let V ( G ) = { v 1 , v 2 , … , v n } . For convenience, for each i ∈ { 1 , … , n } , N ( v i ) = { v i 1 , v i 2 , … , v i t i } , where t i = d G ( v i ) . Furthermore, let d i j = d G ( v i j ) for any j ∈ { 1 , … , t i } . Without loss of generality, for each i ∈ { 1 , … , n } , { v i v i j : 1 ≤ j ≤ l i } is the set of edges incident to v i subdivided exactly once.

Note that if an edge v i v i j in G is subdivided exactly once to be v i x i j v i j in S ( G ) or S 2 ( G ) , the Randić index of the induced path v i x i j v i j in S ( G ) or S 2 ( G ) from itself to its corresponding edge of L ( S ( G ) ) or L ( S 2 ( G ) ) increases by 1 t i d i j and if an edge v i v i j of G is subdivided at least twice ( k ≥ 2 ) to be v i x i j 1 x i j 2 … x i j k v i j of S 2 ( G ) , the Randić index on induced path v i x i j 1 x i j 2 … x i j k v i j of S 2 ( G ) from itself to its corresponding path of L ( S 2 ( G ) ) increases by 1 2 t i − 1 2 . So we may assume those l i edges adjacent to v i are all subdivided only once and t i − l i edges adjacent to v i are all subdivided at least twice. Thus, we consider three cases.

Case 1. l i = 0 , 1 ≤ i ≤ t i .

Since l i = 0 , every edge e in G is subdivided at least twice. So, we have

R ( L ( S 2 ( G ) ) ) − R ( S 2 ( G ) ) = ∑ i = 1 n ∑ 1 ≤ j 1 < j 2 ≤ t i 1 t i − ∑ j = 1 t i 1 2 t i + t i 2 1 2 t i − 1 2 = ∑ i = 1 n t i − 2 t i − 2 4 .

If t i ≥ 6 , then t i − 2 t i − 2 4 ≥ 0 , otherwise t i − 2 t i − 2 4 < 0 . Thus, if δ ( G ) ≥ 6 , we have R ( L ( S 2 ( G ) ) ) > R ( S 2 ( G ) ) .

Case 2. l i = t , 1 ≤ i ≤ t i .

Since l i = t , every edge e in G is subdivided exactly once. So, we have

R ( L ( S ( G ) ) ) − R ( S ( G ) ) = ∑ i = 1 n ∑ 1 ≤ j 1 < j 2 ≤ t i 1 t i − 1 2 ∑ j = 1 t i 1 2 t i + 1 2 ∑ i = 1 t i 1 t i d i j = ∑ i = 1 n 2 t i 2 − 2 t i − t i 3 2 2 2 t i + 1 2 ∑ j = 1 t i 1 t i d i j = ∑ i = 1 n f ( t i ) 2 2 t i + 1 2 ∑ j = 1 t i 1 t i d i j .

The function f ( t ) = 2 t 2 − 2 t − t 3 2 is strictly convex on interval [ 3 , + ∞ ) , since its second derivative f ″ ( x ) = 2 2 − 3 4 t is positive for any t ≥ 3 , implying that f ′ ( t ) ≥ f ′ ( 3 ) > 0 and f ( t ) ≥ f ( 3 ) > 0 . Thus, we have R ( L ( S ( G ) ) ) > R ( S ( G ) ) .

Case 3. 1 < l i < t i , 1 ≤ i ≤ t i .

(5)

(6) R ( L ( S 2 ( G ) ) ) − R ( S 2 ( G ) ) = ∑ i = 1 n t i 2 1 t i − t i 2 t i + 1 2 ∑ j = 1 l i 1 t i d i j + 1 2 ∑ j = l i + 1 t i 1 2 t i − 1 2 ≥ ∑ i = 1 n t i 2 1 t i − t i 2 t i + ∑ j = l i + 1 t i 1 2 t i − 1 2 = 1 2 ∑ i = 1 n l i − 1 − l i ⋅ 2 t i .

Since t i ≥ 3 for any 1 ≤ i ≤ t i , if t i ≥ l i ≥ 6 , then l i − 1 − l i ⋅ 2 t i ≥ 0 in (6), and the results follows.

If t i ≥ l i = 5 for any 1 ≤ i ≤ t i , then by (5),

R ( L ( S 2 ( G ) ) ) − R ( S 2 ( G ) ) ≥ ∑ i = 1 n 1 2 t i ( 4 t i − 5 2 t i ) > 0 ,

implying R ( L ( S 2 ( G ) ) ) ≥ R ( S 2 ( G ) ) .

If t i ≥ l i = 4 for any 1 ≤ i ≤ t i , then by (5),

R ( L ( S 2 ( G ) ) ) − R ( S 2 ( G ) ) ≥ ∑ i = 1 n 1 2 t i ( 3 t i − 4 2 t i ) > 0 .

Therefore, R ( L ( S 2 ( G ) ) ) ≥ R ( S 2 ( G ) ) .

If t i ≥ 4 , l i = 3 for any 1 ≤ i ≤ t i , then by (6),

R ( L ( S 2 ( G ) ) ) − R ( S 2 ( G ) ) ≥ ∑ i = 1 n 1 4 t i + 1 − ( 3 + t i ) 2 t i t i > 0 ,

which implies that R ( L ( S 2 ( G ) ) ) > R ( S 2 ( G ) ) .

If t i ≥ 5 , l i = 2 for any 1 ≤ i ≤ t i , then by (6),

R ( L ( S 2 ( G ) ) ) − R ( S 2 ( G ) ) ≥ ∑ i = 1 n 1 4 t i − 2 2 t i − 2 t i > 0 .

Consequently, R ( L ( S 2 ( G ) ) ) > R ( S 2 ( G ) ) .

If t i ≥ 5 , l i = 1 for any 1 ≤ i ≤ t i , then by (6),

R ( L ( S 2 ( G ) ) ) − R ( S 2 ( G ) ) ≥ ∑ i = 1 n 1 4 t i − 1 − 2 t i − 2 t i > 0 .

Thus, R ( L ( S 2 ( G ) ) ) > R ( S 2 ( G ) ) . This proof is complete.□

Acknowledgments

We would like to thank the referees for their careful reading of the manuscript and for some helpful suggestions that have improved the article.

Funding information: The work was supported by the open project of Key Laboratory in Xinjiang Uygur Autonomous Region of China (2023D04026) and by NSFC (No. 12061073).
Author contributions: All authors contributed equally to the writing of this article and read and approved the final manuscript.
Conflict of interest: The authors declare no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

References

[1] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947), 17–20, DOI: https://doi.org/10.1021/ja01193a005. 10.1021/ja01193a005Search in Google Scholar PubMed

[2] K. C. Das and M. J. Nadjafi-Arani, On maximum Wiener index of trees and graphs with given radius, J. Comb. Optim. 34 (2017), 574–587, DOI: https://doi.org/10.1007/s10878-016-0092-y. 10.1007/s10878-016-0092-ySearch in Google Scholar

[3] Y. Chen, B. Wu, and X. An, Wiener index of graphs with radius two, ISRN Comb. 2013 (2013), 906756, DOI: https://doi.org/10.1155/2013/906756. 10.1155/2013/906756Search in Google Scholar

[4] M. Knor, R. Škrekovski, and A. Tepeh, Digraphs with large maximum Wiener index, Appl. Math. Comput. 284 (2016), 260–267, DOI: https://doi.org/10.1016/j.amc.2016.03.007. 10.1016/j.amc.2016.03.007Search in Google Scholar

[5] M. Knor, R. Škrekovski, and A. Tepeh, Orientations of graphs with maximum Wiener index, Discrete Appl. Math. 211 (2016), 121–129, DOI: https://doi.org/10.1016/j.dam.2016.04.015. 10.1016/j.dam.2016.04.015Search in Google Scholar

[6] M. Knor, R. Škrekovski, and A. Tepeh, Some remarks on the Wiener index of oriented digraphs, Appl. Math. Comput. 273 (2016), 631–636, DOI: https://doi.org/10.1016/j.amc.2015.10.033. 10.1016/j.amc.2015.10.033Search in Google Scholar

[7] T. K. Šumenjak, S. Špacapan, and D. Štesl, A proof of a conjecture on maximum Wiener index of oriented ladder graphs, J. Appl. Math. Comput. 67 (2021), 481–493, DOI: https://doi.org/10.1007/s12190-021-01498-w. 10.1007/s12190-021-01498-wSearch in Google Scholar

[8] M. Knor, R. Škrekovski, and A. Tepeh, Mathematical aspects of Wiener index, Ars Math. Contemp. 11 (2016), no. 2, 327–352, DOI: https://doi.org/10.26493/1855-3974.795.ebf. 10.26493/1855-3974.795.ebfSearch in Google Scholar

[9] S. Spiro, The Wiener index of signed graphs, Appl. Math. Comput. 416 (2022), 126755, DOI: https://doi.org/10.1016/j.amc.2021.126755. 10.1016/j.amc.2021.126755Search in Google Scholar

[10] S. Guo, W. Wang, and C. Wang, Disproof of a conjecture on the minimum Wiener index of signed trees, Appl. Math. Comput. 439 (2023), 127577, DOI: https://doi.org/10.1016/j.amc.2022.127577. 10.1016/j.amc.2022.127577Search in Google Scholar

[11] F. Buckley, Mean distance in line graphs, Congr. Numer. 32 (1981), 153–162. Search in Google Scholar

[12] N. Cohen, D. Dimitrov, R. Krakovski, R. Škrekovski, and V. Vukašinović, On Wiener index of graphs and their line graphs, MATCH Commun. Math. Comput. Chem. 64 (2010), 683–698.Search in Google Scholar

[13] I. Gutman and L. Pavlović, More on distance of line graphs, Graph Theory Notes New York. 33 (1997), 14–18.Search in Google Scholar

[14] V. A. Rasila and A. Vijayakumar, Steiner Wiener index of line graphs, Indian J. Pure Appl. Math. 53 (2022), 932–938, DOI: https://doi.org/10.1007/s13226-021-00199-1. 10.1007/s13226-021-00199-1Search in Google Scholar

[15] B. Wu, Wiener index of line graphs, MATCH Commun. Math. Comput. Chem. 64 (2010), no. 3, 699–706. Search in Google Scholar

[16] M. Randić, On characterization of molecular branching, J. Amer. Chem. Soc. 97 (1975), no. 23, 6609–6615, DOI: https://doi.org/10.1021/ja00856a001. 10.1021/ja00856a001Search in Google Scholar

[17] B. Liu and I. Gutman, Estimating the Zagreb and the general Randić indices, MATCH Commun. Math. Comput. Chem. 57 (2007), 617–632.Search in Google Scholar

[18] B. Zhou and D. Vukičević, On general Randić and general zeroth-order Randić indices, MATCH Commun. Math. Comput. Chem. 62 (2009), 189–196. Search in Google Scholar

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

[20] I. Gutman and J. Tosović, Testing the quality of molecular structure descriptors Vertex-degree based topological indices, J. Serb. Chem. Soc. 78 (2013), no. 6, 805–810, DOI: https://doi.org/10.2298/JSC121002134G. 10.2298/JSC121002134GSearch in Google Scholar

[21] X. Li and I. Gutman, Mathematical Aspects of Randić-Type Molecular Structure Descriptors, Mathematical Chemistry Monographs, Kragujevac, 2006. Search in Google Scholar

[22] X. Li and Y. Shi, A survey on the Randić index, MATCH Commun. Math. Comput. Chem. 59 (2008), 127–156. Search in Google Scholar

[23] M. Randić, D. Plavšić, and N. Lerš, Variable connectivity index for cycle-containing structures, J. Chem. Inf. Comput. Sci. 41 (2001), 657–662, DOI: https://doi.org/10.1021/ci000118z. 10.1021/ci000118zSearch in Google Scholar PubMed

[24] J. A. Rodríguez and J. M. Sigarreta, On Randić index and conditional parameters of a graph, arXiv:1311.7316, 2013, https://doi.org/10.48550/arXiv.1311.7316.Search in Google Scholar

[25] J. A. Rodríguez-Velázquez and J. Tomás-Andreu, On the Randić index of polymeric networks modelled by generalized Sierpinski graphs, MATCH Commun. Math. Comput. Chem. 74 (2015), 145–160. Search in Google Scholar

[26] W. Carballosa, J. M. Rodríguez, J. M. Sigarreta, and N. Vakhania, f-polynomial on some graph operations, Math. 7 (2019), no. 11, 1074, DOI: https://doi.org/10.3390/math7111074. 10.3390/math7111074Search in Google Scholar

[27] B. Bollobás and P. Erdös, Graphs of extremal weights, Ars Combin. 50 (1998), 225–233.Search in Google Scholar

[28] B. Bollobás, P. Erdős, and A. Sarkar, Extremal graphs for weights, Discrete Math. 200 (1999), no. 1–3, 5–19, DOI: https://doi.org/10.1016/S0012-365X(98)00320-3. 10.1016/S0012-365X(98)00320-3Search in Google Scholar

[29] M. Lu, H. Liu, and F. Tian, The connectivity index, MATCH Commun. Math. Comput. Chem. 51 (2004), 149–153. Search in Google Scholar

[30] X. Li and Y. Yang, Sharp bounds for the general Randić index, MATCH Commun. Math. Comput. Chem. 51 (2004), 155–166. Search in Google Scholar

[31] Y. Hu, X. Li, and Y. Yuan, Trees with maximum general Randić index, MATCH Commun. Math. Comput. Chem. 52 (2004), 129–146.Search in Google Scholar

[32] L. H. Clark and J. W. Moon, On the general Randić index for certain families of trees, Ars Combin. 54 (2000), 223–235.Search in Google Scholar

[33] J. Zhang and B. Wu, Randić index of a line graph, Axioms. 11 (2022), no. 5, 210, DOI: https://doi.org/10.3390/axioms11050210. 10.3390/axioms11050210Search in Google Scholar

[34] B. Borovićanin and B. Furtula, On extremal Zagreb indices of trees with given domination number, Appl. Math. Comput. 279 (2016), 208–218, DOI: https://doi.org/10.1016/j.amc.2016.01.017. 10.1016/j.amc.2016.01.017Search in Google Scholar

[35] K. C. Das, On comparing Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 63 (2010), 433–440.Search in Google Scholar

[36] B. Furtula, I. Gutman, and S. Ediz, On diference of Zagreb indices, Discrete Appl. Math. 178 (2014), 83–88, DOI: https://doi.org/10.1016/j.dam.2014.06.011. 10.1016/j.dam.2014.06.011Search in Google Scholar

[37] S. Fajtlowicz, On conjectures of Graffti, Ann. Discrete Math. 38 (1988), 113–118, DOI: https://doi.org/10.1016/S0167-5060(08)70776-3. 10.1016/S0167-5060(08)70776-3Search in Google Scholar

[38] J. M. Rodríguez and J. M. Sigarreta, The harmonic index, in: I. Gutman, B. Furtula, K. C. Das, E. Milovanovic, and I. Milovanovic (Eds.), Mathematical Chemistry Monograph, Vol. 1, University of Kragujevac, Kragujevac, 2017, pp. 229–289. Search in Google Scholar

[39] B. Furtula, I. Gutman, and M. Dehmer, On structure-sensitivity of degree-based topological indices, Appl. Math. Comput. 219 (2013), no. 17, 8973–8978, DOI: https://doi.org/10.1016/j.amc.2013.03.072. 10.1016/j.amc.2013.03.072Search in Google Scholar

[40] H. Whitney, Congruent graphs and the connectivity of graphs, in: J. Eells and D. Toledo, (Eds.), Hassler Whitney Collected Papers. Contemporary Mathematicians, Birkhäuser, Boston, 1992, DOI: https://doi.org/10.1007/978-1-4612-2972-8_4. 10.1007/978-1-4612-2972-8_4Search in Google Scholar

[41] J. Krausz, Démonstration nouvelle dun théorème de Whitney sur les réseaux, Mat. Fiz. Lapok. 50 (1943), 75–85. Search in Google Scholar

[42] F. Harary and R. Z. Norman, Some properties of line digraphs, Rend. Circ. Mat. Palermo (2) 9 (1960), 161–169, DOI: https://doi.org/10.1007/bf02854581. 10.1007/BF02854581Search in Google Scholar

[43] W. Carballosa, A. Granados, D. Pestana, A. Portilla, and J. M. Sigarreta, Relations between some topological indices and the line graph, J. Math. Chem. 58 (2020), 632–646, DOI: https://doi.org//10.1007/s10910-019-01091-4. 10.1007/s10910-019-01091-4Search in Google Scholar

[44] A. Martínez-Pérez and J. M. Rodríguez, Some results on lower bounds for topological indices, J. Math. Chem. 57 (2019), 1472–1495, DOI: https://doi.org/10.1007/s10910-018-00999-7. 10.1007/s10910-018-00999-7Search in Google Scholar

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

[46] D. Pestana, J. M. Sigarreta, and E. Tourís, Geometric-arithmetic index and line graph, J. Math. Chem. 57 (2019), 1427–1447, DOI: https://doi.org/10.1007/s10910-018-00993-z. 10.1007/s10910-018-00993-zSearch in Google Scholar

[47] P. S. Ranjini, V. Lokesha, and I. N. Cangü l, On the Zagreb indices of the line graphs of the subdivision graphs, Appl. Math. Comput. 218 (2011), no. 3, 699–702, DOI: https://doi.org/10.1016/j.amc.2011.03.125. 10.1016/j.amc.2011.03.125Search in Google Scholar

[48] G. Su and L. Xu, Topological indices of the line graph of subdivision graphs and their Schur bounds, Appl. Math. Comput. 253 (2015), 395–401, DOI: https://doi.org/10.1016/j.amc.2014.10.053. 10.1016/j.amc.2014.10.053Search in Google Scholar

[49] I. Gutman and S. J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons, Springer, Berlin, 1989. 10.1007/978-3-642-87143-6Search in Google Scholar

[50] M. F. Nadeem, S. Zafar, and Z. Zaid, On certain topological indices of the line graph of subdivision graphs, Appl. Math. Comput. 271 (2015), 790–794, DOI: https://doi.org/10.1016/j.amc.2015.09.061. 10.1016/j.amc.2015.09.061Search in Google Scholar

Received: 2022-09-14

Revised: 2023-05-20

Accepted: 2023-06-28

Published Online: 2023-08-01

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

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https://doi.org/10.1515/math-2022-0611

Keywords for this article

general Randić index; line graph; Randić index

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