Sharp Bounds for the General Sum-Connectivity Indices of Transformation Graphs

Abstract

Given a graph , the general sum-connectivity index is defined as , where (or ) denotes the degree of vertex (or ) in the graph and is a real number. In this paper, we obtain the sharp bounds for general sum-connectivity indices of several graph transformations, including the semitotal-point graph, semitotal-line graph, total graph, and eight distinct transformation graphs , where .

1. Introduction

In this paper, we consider simple, undirected, and connected graphs. Let be the graph with vertex set and edge set . The order and size of are denoted by and , respectively. For a vertex , denotes the degree of . Two vertices in are adjacent if and only if they are end vertices of an edge, and each of the two vertices is called incident to the edge. Besides, two edges are adjacent to each other if and only if they share a common vertex. The minimum and maximum degrees of graph are denoted by and , respectively. We will use the notations , , and for a path, cycle, and complete graph of order [1], respectively.

The complement of , denoted by , is the graph with and two vertices in are adjacent if and only if they are not adjacent in . Thus, the size of is and if then .

A topological index is a numeric quantity associated with a graph which characterizes the topology of graph. A topological index of a graph is equal to the topological index of , if and only if two graphs and are isomorphic. The idea of topological index appears from work done by Wiener in , this index is called Wiener index. The first and second Zagreb indices have been introduced by Gutman and Trinajestić [2]. These indices are defined on the ground of vertex degrees as follows:The Randić connectivity index was defined in 1975 by Randić [3]. It has been extended to the general Randić connectivity index. The general Randić connectivity index (general product-connectivity index) was defined by Bollobás and Erdős [4] as follows:where is a real number. Then is the classical Randić connectivity index. The sum-connectivity index was proposed in [5]. This concept was extended to the general sum-connectivity index in [6], which is defined aswhere is a real number. Then is the classical sum-connectivity index. The sum-connectivity index and the product-connectivity index correlate well with the -electron energy of benzenoid hydrocarbons [7].

The total graph of the graph is a graph whose vertex set is the union of and such that if and only if and are either adjacent or incident in [8]. Let , , and be the variables having values + or −. The transformation graph is a graph whose vertex set is the union of and , and if and only if(1); then or if and are adjacent or nonadjacent in , respectively;(2); then or if and are adjacent or nonadjacent in , respectively;(3) and ; then or if and are incident or nonincident in , respectively.

There are eight different transformations of the given graph . For instance, is the total graph of with number of vertices and number of edges , and is the complement of total graph . For other transformations of graph, , , and are the complements of , , and , respectively.

The concepts of semitotal-point graph and semitotal-line graph are introduced by Sampathkumar and Chikkodimath [9]. The semitotal-point graph is a graph whose vertex set is the union of and , and if and only if (i) and are adjacent vertices in or (ii) one is a vertex of and the other is an edge of incident to it. Thus, semitotal-point graph has number of vertices and number of edges.

The semitotal-line graph is a graph whose vertex set is the union of and , and if and only if (i) and are adjacent edges in and (ii) one is a vertex of and the other is an edge of incident to it. Thus, semitotal-line graph has number of vertices and number of edges.

Eventually, many properties of these transformation graphs can be determined. For example, the Zagreb indices of transformation graphs and total transformation graphs were calculated by Basavanagoud and Patil [10] and Hosamani and Gutman [11], respectively. Wu and Meng [12] investigated the basic properties (connectedness, graph equations and iteration, and diameter) of total transformation. Xu and Wu [13] determined the connectivity, the Hamiltonian, and the independence number of . Yi and Wu [14] determined the connectivity, the Hamiltonian, and the independence number of .

In this paper, we obtain lower and upper bounds for the general sum-connectivity indices of the above-defined transformation graphs.

2. Main Results

In this section, we discuss the lower and upper bounds for the general sum-connectivity indices of transformation graphs defined in Section 1.

Theorem 1. For , we have , where the equalities hold if and only if is a regular graph.

Proof. Since has vertices and edges, it holds thatNote that if then and if then . It is clear that and Δ. And these equalities hold if and only if is a regular graph. Therefore,Similarly, we can computeThe two equalities in (6) and (7) obviously hold if and only if and are regular, respectively.

Example 2. By Theorem 1, the general sum-connectivity indices of some semitotal-point graphs are given below:(1).(2).(3).

Theorem 3. If then , wherethe equalities hold if and only if is a regular graph.

Proof. Since and , we haveNote that if then and if then . Therefore, we haveSince and , each equality holds if and only if is a regular graph.
After simplification we getSimilarly, we can calculateObviously the equalities in (11) and (12) hold if and only if is a regular graph.

Example 4. By Theorem 3, the general sum-connectivity indices of some semitotal-line graphs are given below:(1).(2).(3).

Theorem 5. Let . Then , wherethe equalities hold if and only if is a regular graph.

Proof. Since and , we haveNote that for and for . SoNote that and . The equalities hold if and only if is a regular graph.
After simplification, we get Similarly, we can compute Since , we can also write the results above asThus, if is a regular graph, then we obtain the equality in (16), (17), and (18).

Example 6. By Theorem 5, the general sum-connectivity indices of some total graphs are given below:(1)(2)(3)

Theorem 7. Let . Then , wherethe equalities hold if and only if is a regular graph.

Proof. For a given graph , since and , then , , and . Using these values, we can compute the required results.

Theorem 8. Let . Then , wherethe equalities hold if and only if is a regular graph.

Proof. Since and , Note that if then and if then Note that and . The equalities hold if and only if is a regular graph. After simplification, we get Similarly, we can compute The equalities in (23) and (24) obviously hold if and only if is a regular graphs.

Theorem 9. Let . Then , wherethe equalities hold if and only if is a regular graph.

Proof. Since and ,Note that for and for . ThenNote that and . The equalities hold if and only if is a regular graph.
After simplification, we getSimilarly, we calculateIf is a regular graph then we obtain the equalities in (28) and (29).

In fully analogous manner, we also arrive at the following.

Theorem 10. If then(1), where(2), where(3), where (4), where In all the above cases, the equalities hold if and only if is a regular graph, respectively.

3. Conclusion

In this paper, we obtain the sharp lower and upper bounds for general sum-connectivity indices of the semitotal-point graph, the semitotal-line graph, the total graph, and the eight distinct transformation graphs , where in terms of the order, minimum degree, and maximum degree of a graph. Moreover, the extremal graphs achieving these bounds have been described.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research is supported by the Fundamental Research Funds for the Central Universities (nos. 2652015193 and 2652017146) and NSFC of China (no. 11701530).