Some Upper Bounds on the First General Zagreb Index

Abstract

The first general Zagreb index or zeroth-order general Randić index of a graph G is defined as where is any nonzero real number, is the degree of the vertex and gives the classical first Zagreb index. The researchers investigated some sharp upper and lower bounds on zeroth-order general Randić index (for ) in terms of connectivity, minimum degree, and independent number. In this paper, we put sharp upper bounds on the first general Zagreb index in terms of independent number, minimum degree, and connectivity for . Furthermore, extremal graphs are also investigated which attained the upper bounds.

1. Introduction

Let be a connected, simple, and finite graph with vertex set and edge set . The total number of elements in is the order of the graph, and the number of edges which are connected with vertex is said to be the degree of the vertex . A vertex is isolated if it has zero degree. For nonadjacent vertices and , a -vertex cut is a subset where and are from different components of and the smallest cardinality set of vertices which separates and is minimal vertex cut. A subset of nonadjacent vertices is called an independent set, and the largest cardinality set among all independent sets of is the independent number. A set is called clique if all the vertices in are adjacent. If a smallest set of vertices exists in a connected graph whose deletion makes it disconnect, then is said to be the vertex connectivity or simple connectivity of .

For a subset , is an induced subgraph of whose vertices are from and edges are with both ends in . A graph is bipartite with such that , , and every edge connects a vertex from to a vertex in . A bipartite graph in which each vertex of is connected with each vertex of by an edge is called a complete bipartite graph. Let and be two vertex disjoint graphs; then, is a graph with the vertex set and the edge set .

A topological index is a number corresponding to a molecule obtained from the molecular structure of the molecule. This number helps to predict the chemical or physical properties of that molecule. Due to strong applications in chemistry and pharmacy, hundreds of degree-based topological indices have been introduced.

The first Zagreb index of a graph , , is defined as [2]

This old and useful topological index helped in obtaining properties of the structure of molecules such as branching, ZE-isomerism, complexity, heterosystems, -electron energy, and many more [3, 4]. The concept of the first general Zagreb index was introduced by Li and Zheng in [5], and it is defined as

where is any nonzero real number. The first general Zagreb index has grabbed attention of many chemists and mathematicians. Liu and Liu discussed some properties of in [6] related to different operations on graph such as edge moving, edge separating, and edge switching. In [7], the authors presented some inequalities involving different graph parameters. In [8], the authors calculated the first general Zagreb index of generalized -sums graphs. We refer the readers to [1, 9–14] for further study about this topological index.

First, we present an auxiliary lemma which is a direct consequence of the definition of the first general Zagreb index [15].

Lemma 1. Let x and y be two nonadjacent vertices of G; then, for , we have

2. Graphs with Given Connectivity and Minimum Degree

In this section, we provide an upper bound on the first general Zagreb index in terms of order, vertex connectivity, and minimum degree. Let be the set of all graphs on vertices, vertex connectivity, and minimum degree, where and .

Theorem 2. Let and , and for , we haveand the equality holds if and only if .

Proof. For , we have ; in other words, . Suppose that and let be the graph in with the maximum first general Zagreb index for . Let be the vertex cut with cardinality . We will prove our result by proving the following claims.
Claim I. consists of exactly two components.

Proof. On the contrary, suppose that consists of at least three components. and are two components of ; then, there will be and such that which is against the assumption of because of Lemma 1. This completes the proof of Claim I.
Now assume that and . Clearly, and where .
Claim II. and are cliques.

Proof. On the contrary, suppose that is not a clique. Then, we have two cases.

Case 1. There are nonadjacent vertices such that , which contradicts the assumption of because of Lemma 1, and we have .

Case 2. Otherwise, joining nonadjacent vertices in will increase the minimum degree of . Then, from the proof of Claim III, we havewhich again contradicts that has the maximal first general Zagreb index because . This completes the proof of Claim II.
From Claim II, for and , we suppose that .
Let ; then, for .This implies that is a decreasing function.
Claim III. We have or .

Proof. On the contrary, suppose that ; then, we have . The last inequality is due to the fact that is a decreasing function for and . This implies that if . This completes the proof of Claim III.
From Claims I, II, and III, we deduce that , which proves the theorem.

3. Bipartite Graphs with Given Connectivity

Let denote the set of bipartite graphs of order and vertex connectivity . Now we introduce a graph obtained from the graph by joining a new vertex to vertices of degree of .

Theorem 3. Letand. Then, for,and the equality holds if and only if eitheror, where

Proof. Note that , so we consider . Let be the graph with the maximum first general Zagreb index having a -vertex cut set . Let such that . Furthermore, we have and . The required result is obtained by proving the following claims.
Claim I. and are complete bipartite graphs, where is one of the components in .

Proof. Suppose on the contrary that or is not a complete bipartite graph. Then, there are two nonadjacent vertices in and . From Lemma 1, we know that which is against the maximality of . Hence, and are complete bipartite graphs.
Claim II. If and are nonempty subsets of , then has exactly two components.

Proof. has at least three components and and are two of these components. Then, there are two vertices such that with being a -vertex cut of . By Lemma 1, we have a contradiction that has the maximum first general Zagreb index.
Claim III. Either is empty or is empty.

Proof. On the contrary, suppose that and are nonempty sets; then, by Claim II, has exactly two components named and . Let and . Assume that and .(i)Now we construct a new graph from as . By the definition of the first general Zagreb index, we have , and the last inequality can be seen by considering the function . is an increasing function for and , and we have ; this implies that .(ii)Let and consider arbitrary vertices . Now we construct a new graph from by adding more edges between the vertices of and and adding further edges . Clearly, is the vertex cut with cardinality ; in other words, . From Lemma 1 and of Claim III, we have , which is a contradiction.From above claim, we deduce that is the -vertex cut of .
Claim IV. consists of an isolated vertex.

Proof. On the contrary, suppose that the components of are complete bipartite graphs. Also, suppose that and , where for .
Without loss of generality, suppose that and . Construct a new graph from as . This implies that is also a -vertex cut of and . Similar to Claim III , we have , which is against the maximality of the first general Zagreb index of .
By definition of the first general Zagreb index, we have

4. Graphs with Given Connectivity and Independent Number

Let be the set of graphs of order , vertex connectivity , and independent number . In this section, we investigate the graph which gives the maximum general first Zagreb index from .

Theorem 4. Let with and . Then, for , we haveand the equality holds if and only if .

Proof. For , this result has been discussed in [16]. So, we assume that , and let be the graph with the maximum first general Zagreb index in and be the -vertex cut and maximum independent sets of , respectively. The following claims will prove our main result.
Claim I. Let be one component of and ; then, we have and .

Proof. As has the maximum first general Zagreb index and by Lemma 1, we have . On the contrary, suppose that ; this implies that we have vertices and such that or . Furthermore, one can notice that . By Lemma 1, we have a contradiction of the choice of , which proves the claim.
Claim II. contains exactly two components.

Proof. On the contrary, suppose that contains at least three components, and two of them are named as and . Let . Then, , and we have a contradiction on the maximality of by Lemma 1.
Claim III. If , then .

Proof. of Claim III. On the contrary, we suppose . Furthermore, if , then as is a connected component. Suppose that ; then, . Otherwise, , and choosing , we have that is an independent set which is a contradiction to the definition of .
Let , and we construct a new graph as . Clearly, is the maximal independent set, and is the minimal vertex cut set of ; .
Let and ; then we have . By applying the definition of on and , we obtain which is against the choice of .
From Claims I, II, and III, we have , where is an isolated vertex of . Let ; then,Let . Consider the function for . As , is a decreasing function for . This implies that , and the last inequality is due to Jensen’s inequality. Hence, (1) attains its maximum value for ; in other words, for , the first general Zagreb index attains its maximum value for .

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.