Abstract
Let be a connected graph. The resistance distance between two vertices and in , denoted by , is the effective resistance between them if each edge of is assumed to be a unit resistor. The degree resistance distance of is defined as , where is the degree of a vertex in and is the resistance distance between and in . A bicyclic graph is a connected graph with . This paper completely characterizes the graphs with the second-maximum and third-maximum degree resistance distance among all bicyclic graphs with vertices.
1. Introduction
All graphs considered in this paper are simple and undirected. Let be a graph with vertices and edges. Let be the set of vertices adjacent to in . The degree of in , denoted by , is equal to . Denote the minimum degree of vertices in by . A vertex of degree one is called a pendant vertex, and the edge incident with a pendant vertex is called a pendant edge. The distance between two vertices and of , denoted by or , is the length of a shortest path connecting and in . For a subset of , denote by , the subgraph induced by and the graph . We use instead of if for simplicity. Let and be the path and the cycle graphs on vertices, respectively.
A topological index or a graph-theoretic index is a real number related to a graph. Topological indices of molecular graphs are one of the oldest and most widely used descriptors in quantitative structure-activity relationships [1, 2]. One of the most exhaustively studied [3, 4] topological indices is the Wiener index. The Wiener index was introduced in 1947 [5] and defined as . It is well correlated with many physical and chemical properties of organic molecules and chemical compounds.
Based on the electrical network theory, Klein and Randić [6] proposed a novel distance function called resistance distance in 1993. They treated a graph as an electric network by considering each edge of as a unit resistor. Then, the resistance distance between two vertices and in , denoted by , is defined as the effective resistance between them. Klein and Randić [6] also proved that , with equality if and only if there is a unique path connecting and in . In recent years, this new type of distance between vertices in a graph has attracted prominent attention in mathematics and chemistry [6–11].
Similar to the Wiener index, the Kirchhoff index of a graph is defined as
This invariant has wide applications in electric circuit, physical interpretations, chemistry, and graph theory [12–16].
In 2012, Gutman et al. [17] introduced the concept of the degree resistance distance defined as
Palacios called it as additive degree-Kirchhoff index in [18]. In [17], Gutman et al. [17] presented some properties of and characterized the unicyclic graphs with the minimum and second-minimum . Later, the unicyclic graphs with the maximum and second-maximum -value were considered in [19, 20]. In [21, 22], the cactus graphs with the minimum, the second-minimum, and the third-minimum -values were also completely characterized. Recently, the bicyclic graphs with maximum and minimum -values were determined in [23, 24], respectively.
A bicyclic graph is a connected graph such that . The kernel of , denoted by , is the unique bicyclic subgraph of with no pendant vertices. Any bicyclic graph is obtained from its kernel by attaching trees to some vertices in . Given a family of graphs , the graphs with the maximum and second maximum values of topological indices among are examined widely, see in [25–29]. Motivated by this, in this paper, we determine the graphs with the second-maximum and third-maximum degree resistance distance among all bicyclic graphs with vertices.
2. Preliminaries
Let be the set of bicyclic graphs of order , be the set of bicyclic graphs of order with exactly two cycles, and . Let be obtained from two vertex-disjoint cycles and by identifying a vertex and a vertex , be obtained from two vertex-disjoint cycles and by connecting a vertex and a vertex by a path of length , and be the union of three internally disjoint paths , , and , respectively, with common end vertices, where and at most one of them is 2.
Let be a graph and be a vertex in . Define and .
We present a few lemmas which will be employed later to establish our main results.
Lemma 1 (see [13]). Let be a connected graph with a pendant vertex with its unique neighbor . Then, .
Lemma 2 (see [13]). Let be a bicyclic graph of order and . Then, . Moreover, if , then .
The following remark can be obtained from the proof of Lemma 2.
Remark 1. Let be a graph in and . Then, .
Lemma 3 (see [17]). Let be a connected graph with a cut vertex such that and are two connected subgraphs of having as the only common vertex and . Let . Then, .
Lemma 4 (see [17]). Let be a cycle with length and . Then, ,, , and .
Lemma 5 (see [23]). Let be a connected graph of order and be a cycle of order . Let be the graph of order obtained from by attaching one pendant path of order to one vertex of . Further suppose is the graph obtained from and by identifying one vertex in and one vertex in ; is the graph obtained from and by identifying one vertex in and the pendant vertex in . Then, we have .
By an argument similar to that of Lemma 5, we easily get the following result.
Lemma 6. Let be a connected graph of order and be a cycle of order . Let be obtained by identifying a pendant vertex of with any vertex of . Suppose is the graph obtained from and by identifying one vertex in and one vertex in ; is obtained from and by identifying one vertex in and the pendant vertex in . Then, .
In [23], Du and Tu characterized the unique bicyclic graph with maximum degree resistance distance. They also presented two significant lemmas in [23].
Theorem 1 (see [23]). Let be a bicyclic graph of order ; then, , with equality if and only if .
Lemma 7 (see [23]). Let be a bicyclic graph of order and . Then, .
Lemma 8 (see [23]). Let be a bicyclic graph of order , be a pendant vertex of , and be its neighbor. Then, .
3. Bicyclic Graphs with the Second-Maximum Degree Resistance Distance
In this section, we will determine the bicyclic graphs with the second-maximum degree resistance distance.
Suppose . Let be obtained from two 3-cycles and by connecting and by a path . Define and , where . Let be obtained from a 4-cycle and a path by adding the edges (, resp.) and . Let be obtained from a 4-cycle and a 3-cycle by connecting and by a path (see Figure 1). Then, we have the following lemma.
Lemma 9. Let , , , , and be defined as above. Then, ,, , , and .
Proof. By Lemma 8 and Theorem 1, we easily obtainLet . By Lemma 3,Let . By Lemmas 3 and 6,Let . By Lemma 3,
Theorem 2. Suppose is a graph in with and . Then, , with equality if and only if , where is defined as in Lemma 9.
Proof. It is easy to verify that, for any graph in with , , with equality if and only if .
Now, we assume and consider the following two cases. Case 1 δ(G) = 1: let be a pendant vertex in . If , then either , or , where and are defined as in Lemma 9. By Lemma 9, If , we prove it by induction on . Let be the neighbor of . By the inductive hypothesis, Remark 1, and Lemmas 7–9 Case 2 (): in this case, is of the form or . By Lemmas 5 and 6, we have , with equality if and only if . Note that by Lemma 9. Therefore, the proof is complete.
Theorem 3. Suppose is a graph of order in. Then, , with equality if and only if , where is defined in Lemma 9.
Proof. It is easy to verify that the only graph in is and . We assume next, and consider the following two cases. Case 1 (): let be a pendant vertex in and be the neighbor of . We prove it by induction on . By the inductive hypothesis, Lemma 2, and Lemmas 7–9, The equality holds if and only if , , and . By the inductive hypothesis, , which is obtained from a 4-cycle and a path by adding the edges and . We show that , i.e., . By direct calculation, we have , , and . Obviously, if . Therefore, , i.e., . Case 2 (): then, is of the form . Suppose and are the only two vertices of degree 3. Since (see [13]), we have If , then . For any graph when , we have calculated and found that .Combining Theorems 1–3, we can obtain the first main result of our paper.
Theorem 4. Suppose is a bicyclic graph of order with. Then,, with equality if and only if , where is defined as in Lemma 9.
4. Bicyclic Graphs with the Third-Maximum Degree Resistance Distance
In this section, we will determine the bicyclic graphs with the third-maximum degree resistance distance.
Lemma 10. Let be obtained from a 4-cycle, a pathand an isolated vertexby adding the edges,, and, whereand. Then,,,, and, for .
Proof. By Lemmas 8 and 9, we easily obtainand for ,
Proposition 1. Suppose is a bicyclic graph of order and , where is defined as in Lemma 9. Then, .
Proof. It is not hard to verify that, for any bicyclic graph of order 5 and , . Thus, we assume in the following cases. Case 1 (): let be the neighbor of . Suppose , where is obtained from a 4-cycle and a path by adding the edges and . Then, since . By Lemma 1, If , we shall prove it by induction on . By the inductive hypothesis, . Case 2: .By Lemma 2, .
Lemma 11 (see [23]). Let be a bicyclic graph of order , be a pendant vertex of , and be its neighbor. Then, .
Proposition 2. Let be a graph in of order and , where is defined as in Lemma 9. Then, .
Proof. It is easy to verify that for any graph with and , . Thus, we assume in the following cases. Case 1 (): let be the neighbor of . Suppose , where is obtained from a 4-cycle and a path by adding the edges and . Then, since . Moreover, by Lemma 11. By direct calculation, we get , , , and if . Thus, and . If , we prove it by induction on . By the inductive hypothesis, . Case 2: . Subcase 1: is not contained by any cycle of . By the same argument as that of Case 2 of Lemma 2.6 in [23], we can construct a series of bicyclic graphs in such that and is a pendant vertex in , where . Suppose . Then, is obtained from a 4-cycle and a path by adding the edges and . By the transformation from to , we can conclude that , i.e., . Note that . We have . If , then, by Case 1, . Subcase 2: is in a cycle of .Let be the kernel of . By Claims 1 and 2 of Lemma 2.6 in [23], we can construct a graph in having as its kernel and . Moreover, is obtained from by attaching a pendant path to the vertex , where is a vertex of such that .
Suppose has only two vertices of degree three, say and . Without loss of generality, we assume that , and . Then, by Lemma 2,Suppose has exactly three vertices of degree three, say , , and . Let be the pendant vertex of . Without loss of generality, we assume that . Then, by Lemma 2,Suppose has a vertex of degree four, say , and a vertex of degree three, say . Let be the pendant vertex of . Without loss of generality, we assume that . Then, by Lemma 2,which completes the proof.
Theorem 5. Suppose is a graph of order in. Then, , with equality if and only if , where and are defined as in Lemma 9.
Proof. It is not hard to verify that, for any graph in , , with equality if and only if .
We assume that , and consider the following two cases. Case 1: . Let be a pendant vertex of . Suppose , where is obtained from a 4-cycle , and a path by adding the edges and . Then, , where , and is defined in the Lemmas 10. By Lemma 10, for . If , we prove it by induction on . Let be the neighbor of . By the inductive hypothesis, Lemma 8, and Propositions 1 and 2, The equality holds if and only if , , and . By the inductive hypothesis, , where is obtained from a 4-cycle and a path by adding the edges and . We show that , i.e., . By direct calculation, we have , , and . Obviously, if . Therefore, , i.e., . Case 2: .By a similar argument to that of Case 2 in Theorem 3, we obtainIf , then . For any graph of the form when , we have calculated and found that .
From Theorems 2 and 4, we obtain the following result.
Theorem 6. Letandbe defined as in Lemma 9. Then, among all bicyclic graphs of order ,(i)If , the graph is the unique graph with the third-maximum degree resistance distance of value (ii)If , the graph is the unique graph with the third-maximum degree resistance distance of value
5. Conclusion
As a molecular structure descriptor, the Wiener index is one of the widely employed topological indices, as it is well correlated with many physical and chemical properties of a variety of classes of chemical compounds. A weighted version of the Wiener index is the degree resistance distance. In this paper, we characterize the graphs with the second-maximum and third-maximum degree resistance distance among all bicyclic graphs with fixed order. Furthermore, we present an open problem.
Problem 1. Characterize the tricyclic graphs of order with the maximum and second-maximum degree resistance distance.
Data Availability
All the proofs and exemplary data of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The research of the first author was supported by National Natural Science Foundation of China (no. 11801568).