Journal of Chemistry

Volume 2017 (2017), Article ID 3531746, 7 pages

https://doi.org/10.1155/2017/3531746

## On the Resistance-Harary Index of Graphs Given Cut Edges

Huaiyin Institute of Technology, Faculty of Mathematics and Physics, Huai’an, Jiangsu 223003, China

Correspondence should be addressed to Hongzhuan Wang

Received 22 June 2017; Accepted 26 November 2017; Published 21 December 2017

Academic Editor: Robert Zaleśny

Copyright © 2017 Hongzhuan Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Graphs are often used to describe the structure of compounds and drugs. Each vertex in the graph represents the molecule and each edge represents the bond between the atoms. The resistance distance between any two vertices is equal to the resistance between the two points of an electrical network. The Resistance-Harary index is defined as the sum of reciprocals of resistance distances between all pairs of vertices. In this paper, the extremal graphs with maximum Resistance-Harary index are determined in connected graphs with given vertices and cut edges.

#### 1. Introduction

Recently, the development of computational chemistry owes much to the theory of graphs. One of the most popular areas is topological index. The molecular topological index can describe the molecular structure quantitatively and analyze the structure and performance of molecules.

Among them, the most common topological index is resistance distance. The resistance distance is raised by Klein and Randić [1] as a distance function. Let be a simple connected graph with vertex set and edge set .

The resistance distance between vertices and of is recorded as , which represents the effective resistance between two nodes and in an electronic network; that is to say, the vertex corresponds to the node of the electronic network, and the edge corresponds to the unit resistance.

Similar to the traditional path distance, the resistance distance not only has good mathematical characteristics but also has good physical characteristics [2, 3]; at the same time, it also has a good application in chemistry.

Harary index is another kind of graph invariants proposed by Plavšić et al. [4] and by Ivanciuc et al. [5] in 1993 for the characterization of molecular graphs. Name this in honor of Professor Frank Harary’s birthday. The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of the graph ; that is,

Gutman [6] and Xu [7] investigated the Harary index of trees; they studied the Harary index of tree and pointed out that the path and the star attain the minimal and maximal value of Harary index, respectively, among a tree with given n vertices. In recent years, the Harary index was well studied in mathematical and chemical literatures [8].

The reciprocal resistance distance is also called electrical conductance, Klein and Ivanciuc [9] investigated QSAR and QSPR molecular descriptors computed from the resistance distance and electrical conductance matrices, and they proposed the global cyclicity index as where the sum is over all edges of .

In [10], using graph theory, electronic networks, and real number analysis methods, Yang obtains some conclusions about the global cyclicity index.

Following the definition of the Harary index, Chen et al. [11] generalized the global cyclicity index and introduced Resistance-Harary index, defined as

In [11], Chen et al. depicted the graphs with largest and smallest Resistance-Harary index in all unicyclic graphs.

All graphs considered in this paper are finite and simple.

Before proceeding, we introduce some further notation and terminology. A cut edge is an edge whose deletion increases the number of components. Denote by , , and the complete, cycle, and star graph on vertices, respectively. For a graph with , denotes the graph resulting from by deleting and its incident edges. For an edge of the graph (the complement of , resp.), (, resp.) denotes the graph resulting from by deleting (adding, resp.) .

For other definitions, we can refer to [12].

In this thesis, we consider the Resistance-Harary index of graphs given cut edge. We will determine the graphs with maximum Resistance-Harary index in connected graphs given vertices and cut edges.

#### 2. Some Preliminary Results

We first list or prove some lemmas as basic but necessary preliminaries.

Lemma 1 (see [1, 13, 14]). *Let be a cut vertex of a graph , and let and be vertices in different components of . Then *

Lemma 2. *The functions for and and for and , are strictly increasing.*

*Proof. *By simple calculation, Since and , is the derivative of function on . Obviously, is a strictly increasing function for and .

Similarly, we prove that the function for and , is also an increasing function. We finally obtain the result.

Lemma 3. *Let be a connected graph with at least three vertices. If is not isomorphic to , Let , and then .*

*Proof. *Suppose that is not a complete graph. Then there exists a pair of vertices and in such that .

Let , we have We only to prove that ; the proof of for is similar. We distinguish the following two cases.*Case 1*. and are vertices of cycle in , where is the length of .

Let and be distance between and in the cycle , respectively. By the definition of resistance, we have That is to say, .*Case 2*. and are not vertices in any cycle of .

In this case, we have This completes the proof.

Lemma 4. *Let be a cut edge in , and let and be the two components of . Suppose further that and , and then .*

*Proof. *By the definition of Resistance-Harary index and by Lemma 1, we have This completes the proof.

Lemma 5. *Let and be the graphs in Figure 1, where is a complete graph, and then , with equality if and only if .*