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, 9–17].
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 [20–31]). 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 [32–34].
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).
(a)
(b)
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 .
(a)
(b)
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 have Thus, Thus, Similarly, Then,(iii)Note that , for in and for in ; then, we have(iv)Note that , for in , and for in ; then, we have Thus, 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 .
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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 , for any in and in ; then, we have(iii)Note that , , , for in and , respectively; then, we have(iv)Note that for and for ; then, we have Combing , we have(v)Note that , and , for in and , respectively; then, we have(vi)Note that for and for ; then, we have(vii)Note that for , and , for , ; by the symmetry between and , we have Since , if , then . Then, Thus,(viii)Note that , for and , for ; then, we have Then, When , ; thus, and Since , , if , then . Then, Thus, When , combing , we have Let ; this function is strictly increasing in this interval; thus, . Hence,(ix)Note that , for in and , for in ; then, we haveThus, by (i)–(ix), when , we get .
When , . By comparison of direct calculation, we can obtain .
The proof is completed.
3. Maximum Reciprocal Degree Resistance Distance Index of Unicyclic Graphs
Let be a connected graph with exactly one cycle, say . can be viewed as a graph obtained by coalescing with a number of trees, denoted by , by identifying with some vertex of for all . Denote by the unicyclic graph obtained from cycle by attaching pendant edges to a vertex of . The next theorem discusses the maximum reciprocal degree resistance distance index in , which is the main result of our work.
Theorem 1. Let be a unicyclic graph of order ; then, , and the equality holds if and only if .
Proof. Let be a unicyclic graph of order such that is as big as possible. By Lemmas 2–4, we have ; the equality holds if and only if .
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This study was jointly supported by the National Natural Science Foundation of China (11871077), the Natural Science Foundation of Anhui Province (1808085MA04), and the Natural Science Foundation of Department of Education of Anhui Province (KJ2017A362).