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 bicyclic graphs without common edges and with vertices. We study the graph with the 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 vertex, , denoted by the degree of vertex . The distance between vertices of and is defined as the length of the shortest path between them and denoted by . The resistance distance between vertices of and , denoted by , is the effective resistance between the two nodes of the electronic network obtained so that its nodes correspond to the vertices of and each edge of is replaced by a resistor of unit resistance.
Topological indices are numbers related to molecular structures, used as quantitative relationships between chemical structures and properties. Some of them are based on the distance of graph [1], the vertex degree [2], or the resistance distance. The first topological index was published by Wiener [3]. From then, many topological indices are defined, such as the Harary index [4], Kirchhoff index [5, 6], the first and the second Zagreb indices [7, 8], and degree distance [9, 10].
Recently, Alizadeh and Iranmanesh [11] proposed a new topological index, the reciprocal degree distance, which is defined as
Its lower and upper bounds were obtained in [12], and its chemical applications and mathematical properties have been well studied in [11, 13].
Especially, analogous to the reciprocal degree distance index , Cai et al. [14] introduced a new graph invariant based on both the vertex degree and the resistance, named the reciprocal degree resistance distance index, shown as follows:
A bicyclic (or unicyclic) graph is a connected graph of order with (or ) edges. Let be the set of bicyclic graphs without common edges of order . Tree, unicyclic graph, and bicyclic graph are three kinds of graphs with a simple structure; the reciprocal degree resistance distance index of tree is the same as the reciprocal degree distance, Cai et al. [14] determined the graph with the maximum reciprocal degree resistance distance index among all unicyclic graph and characterized the corresponding extremal graph. In this paper, we determine the graph with the maximum reciprocal degree resistance distance index among all graphs in and characterize the corresponding extremal graph.
This paper is organized as follows: in the second part, we give three types of transformation, edge-lifting transformation, cycle-lifting transformation, and cycle-shrinking transformation, to keep the reciprocal degree resistance distance index increasing. In the third part, we give the maximum reciprocal degree resistance distance index among all graphs in .
2. Preliminaries
Let be the cycle graph with girth . For any two vertices with , by Ohm’s law, one has
Lemma 1. (See [6]). Let be a cut vertex of a connected graph and and be two vertices occurring in different components which arise upon the deletion of .Then,
2.1. Edge-Lifting Transformation
Let be obtained from by an edge-lifting transformation at , see Figure 1. See [14] for specific description.
Lemma 2. (See [14]). If can be obtained from by an edge-lifting transformation, see Figure 1, then .
2.2. Cycle-Lifting Transformation
Let be the star graph with order and be the star graph with order . Let be a bicyclic graph with exactly two cycles, which are and , and is obtained from by identifying with the center of for all and with the center of for all . Deleting all edges in , joining to all pendent 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, see Figure 2, and , then .
Proof. Let and be as shown in Figure 2. By the definition of ,(1)Noting that , for of and of , we have(2)Noting that , for and for of , and for of , we have(3)Noting that , for of , , , for of , and , for of , we have(4)Noting that , for of and of , similarly for and of and , respectively, and for and of and , respectively, we have the same conclusions; thus,presents.(5)We note that , for any of and , for any of . Let , when ; then, . Thus, we haveand then, we have Thus, Similarly, Then,(6)It is noted that , , for and of . Similarly, It is noted that , , , for and of . Let , when . Let , when . Then, for and . Thus, we haveand then, we have Similarly,and then, we have Thus,(7)For of and , respectively, simulating the abovementioned calculation method of (6), we get the similar conclusion,(8)Obviously,
Thus, combing (1)–(8), we get , and the results are as follows.
Corollary 1. If can be obtained from by a cycle-lifting transformation, see Figure 3, and , then .
2.3. Cycle-Shrinking Transformation
Let , which is obtained by coalescing and at the vertex and attaching pendent edges to the vertex . We note that . Deleting the edges and adding the edges , we get This operation is called cycle-shrinking transformation. It is denoted by (, resp), the set of pendent vertices of (, resp). Let ; can be partitioned into two subsets: one has vertices, which is naturally corresponding to (also denoted by ), and another has vertices, denoted by .
Lemma 4. Let ; if can be obtained from by a cycle-shrinking transformation, see Figure 4, then .
Proof. Let and be as shown in Figure 4. By the definition of ,(1)Noting that , for any of and , respectively, we have(2)Noting that , , for in and , respectively, we have(3)Noting that , for any in and in , we have(4)Noting that for and for , we have Thus, combing (2) and (4), we have(5)Noting that , for any of and , respectively, we have(6)Noting that and for of and , respectively, we have(7)Noting that for of and for of , we have(8)Noting that for of and , respectively, we have(9)Noting that , and , , we have(10)Noting that for , of and , for of , by the symmetry between and , we have Since , if , then , and Thus,(11)Noting that for , of , for of , and , by symmetry between and , we have(12)Noting that , for of and , for of , we have When , we denote , and then, ; thus, Thus, Since , . If , then , and Thus, When , combing (6), (7), (10), and (12), we have(i)Let ; this function is strictly increasing in this interval; thus, . Thus,(13)Noting that , for of and , for of , we have
Thus, combining (8) and (13), we have
So, if , we get , and the results are as follows.
If , . By comparison of direct calculation, we can obtain .
The proof is completed.
Corollary 2. If can be obtained from by a cycle-shrinking transformation, see Figure 5, and , then .
3. Maximum Reciprocal Degree Resistance Distance Index of Bicyclic Graphs
Let ; applying a sequence of edge-lifting transformations, cycle-lifting transformations, and cycle-shrinking transformations to , we can obtain a new graph , denoted by , see Figure 5. The next theorem discusses the maximum reciprocal degree resistance distance index in .
Theorem 1. Let ; then, , with equality holding if and only if .
Proof. Let such that the is as big as possible. By Lemma 2–4, Corollary 2, we have , and equality holds if and only if .
4. Conclusions
In this paper, we characterize the maximum reciprocal degree resistance distance index among all bicyclic graphs without common edges of order by using three types of transformation which keep the reciprocal degree resistance distance index increasing: edge-lifting transformation, cycle-lifting transformation, and cycle-shrinking transformation.
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 work was jointly supported by the National Natural Science Foundation of China (11871077), the Natural Science Foundation of Anhui Province (1808085MA04 and 1908085MC62), and the Natural Science Foundation of Department of Education of Anhui Province (KJ2017A362).