Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2020 / Article
Special Issue

Mathematical Models and Computation in Discrete Dynamics

View this Special Issue

Research Article | Open Access

Volume 2020 |Article ID 7318968 | https://doi.org/10.1155/2020/7318968

Gai-Xiang Cai, Xing-Xing Li, Gui-Dong Yu, "Maximum Reciprocal Degree Resistance Distance Index of Unicyclic Graphs", Discrete Dynamics in Nature and Society, vol. 2020, Article ID 7318968, 14 pages, 2020. https://doi.org/10.1155/2020/7318968

Maximum Reciprocal Degree Resistance Distance Index of Unicyclic Graphs

Guest Editor: Muhammad Javaid
Received07 May 2020
Accepted26 Jun 2020
Published08 Aug 2020

Abstract

The reciprocal degree resistance distance index of a connected graph is defined as , where is the resistance distance between vertices and in . Let denote the set of unicyclic graphs with vertices. We study the graph with maximum reciprocal degree resistance distance index among all graphs in and characterize the corresponding extremal graph.

1. Introduction

Let be a simple connected graph of order with vertex set and edge set . For any , is the degree of vertex , and the distance between vertices of and , denoted by , is the length of a shortest path between them. Topological indices are numbers associated with molecular structures which serve for quantitative relationships between chemical structures and properties. The first such index was published by Wiener [1], but the name topological index was invented by Hosoya [2]. Many of them are based on the graph distance [3] and the vertex degree [4]. In addition, several graph invariants are based on both the vertex degree and the graph distance [5].

One of the most intensively studied topological indices is the Wiener index. The Wiener index was introduced by American chemist Wiener in [1], defined as

Another distance-based graph invariant Harary index has been introduced by Plavšić et al. [6] and independently by Ivanciuc et al. [7] in 1993 for the characterization of molecular graphs. The Harary index of graph is defined as

For more results related to Harary index, refer to [8, 917].

The resistance distance (if more than one graphs are considered, we write in order to avoid confusion) between vertices and in is defined as the effective resistance between the two nodes of the electronic network obtained so that its nodes correspond to the vertices of and each edges of is replaced by a resistor of unit resistance, which is compared by the methods of the theory of resistive electrical networks based on Ohm’s and Kirchoff’s laws.

The Kirchhoff index of a graph is defined as [18, 19]

As a new structure descriptor, the Kirchoff index is well studied (see recent papers [2031]). In 2017, Chen et al. [32] introduced a new graph invariant reciprocal to Kirchoff index, named Resistance-Harary index:

For more results related to Resistance-Harary index, refer to [3234].

The first and the second Zagreb indices are defined asrespectively. These are the oldest [35, 36] and best studied degree-based topological indices (see the reviews [4], recent papers [37, 38], and the references cited therein).

Dobrynin and Kochtova [39] and Gutman [40] independently proposed a vertex-degree-weighted version of Wiener index called degree distance, which is defined for a connected graph G as

The reciprocal degree distance [41] is defined as

Hua and Zhang [42] have obtained lower and upper bounds for the reciprocal degree distance of graph in terms of other graph invariants. The chemical applications and mathematical properties of the reciprocal degree distance are well studied in [41, 43].

Analogous to the relationship between degree distance and reciprocal degree distance, we introduce here a new graph invariant based on both the vertex degree and the graph distance, named the reciprocal degree resistance distance index:

A unicyclic graph is a connected graph with vertices and edges. Let denote the set of unicyclic graphs with vertices. In this paper, we determine the graph with maximum reciprocal degree resistance distance index among all graphs in and characterize the corresponding extremal graph.

2. Preliminaries

In this section, we will introduce some useful lemmas and three transformations.

Let be the cycle on vertices. For any two vertices with , by Ohm’s law, one has

Lemma 1 (see [19]). Let be a cut vertex of a connected graph and let and be vertices occurring in different components which arise upon deletion of . Then,

2.1. Edge-Lifting Transformation

Let and be two graphs with and vertices, respectively. is the graph obtained from and by adding an edge between a vertex of to a vertex of . And is the graph obtained by identifying of to a vertex of and adding a pendant edge to . We say that is obtained from by an edge-lifting transformation at (see Figure 1).

Lemma 2. If can be obtained from by an edge-lifting transformation, then .

Proof. Consider and shown in Figure 1. By the definition of ,(i)Note that , for or , and , ; then, we have(ii)Note that , for ; then, we have(iii)Note that for any , and ; then, we have(iv)Note that , for any , and ; then, we haveThus, by (i)–(iv), we get .

2.2. Cycle-Lifting Transformation

Let be a graph as shown in Figure 2. Take a cycle in , say ; can be viewed as a graph obtained by coalescing with a number of star subgraphs of , say , by identifying with the center of for all , denoted as and . Deleting all edges in and joining to all pendant vertices of , we obtained a new graph, denoted by (see Figure 2). This operation is called a cycle-lifting transformation of with respect to .

Lemma 3. If can be obtained from by a cycle-lifting transformation, then .

Proof. Consider and shown in Figure 2. By the definition of ,(i)Note that , for in , and in ; then, we have(ii)Let ; when , then . for any in and for in ; then, we haveThus,Thus,Similarly,Then,(iii)Note that , for in and for in ; then, we have(iv)Note that , for in , and for in ; then, we haveThus, we have(v)Note that , for any in and for in ; when , then ; we haveThus,and we haveThus,Similarly,Thus, we haveThen,Thus, by (i)–(v), we get .

2.3. Cycle-Shrinking Transformation

Denote by the unicyclic graph obtained from cycle by attaching pendant edges to a vertex of (see Figure 3). Let ; deleting the edges and adding the edges , we obtain the graph . This operation is called cycle-shrinking transformation. Denote by the set of pendant vertices of . Let , . It is clear that and . Then, ; thus, can be partitioned into two subsets. One has vertices, which is naturally corresponding to . So, we also denote it by ; another has vertices, denoted by .

Lemma 4. Let be an unicyclic graph of order and a cycle with ; if can be obtained from by a cycle-shrinking transformation, then .

Proof. Consider and shown in Figure 3. By the definition of ,(i)Note that , for any ; then, we have(ii)Note that ,