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 .
Proof. From the definition of the Resistance-Harary index and Lemma 4, we have Similarly, Since is a complete graph, for . And with equality if and only if ; that is, . This completes the proof.
Lemma 6. Let and be the graphs depicted in Figure 2, where and are all complete graphs. Let be the number of vertices of , where . If and may be an empty set, then .
Proof. By the definition of the Resistance-Harary index and Lemmas 1 and 4, we have Since and are all complete graphs, for any two vertices , we have , where . Similarly, we have We just need to prove that In fact, In the following, we just need to prove that the fraction of is greater than zero. Denote to be the fraction of ; in fact, Since , . This completes the proof.
3. Characterization of the Maximizing Graph
In this section, we will characterize maximizing graph among all connected graphs with vertices and cut edges.
First, we need some definitions below; let be a star with vertex set , where is the center of the star. In particular, the graph with is obtained from by replacing the vertex by for . By the definition of Resistance-Harary index, it is not difficult to obtain . If is a complete graph, for any vertices , there is .
Lemma 7. Let and be positive integers such that , , and . Let (see Figure 3) and . Then .
Proof. By direct calculation, we have Similarly, we can deduce the value of . Then In order to prove that the result , we distinguish three steps in the following. Denote Obviously, . In the following, we first prove that . By direct calculation, we have (since ). Secondly, we will prove that . where the second inequality is due to the fact thatwhere , , and the function is an increasing function, and then By Lemma 2, the last inequality is due to the fact the function for and is an increasing function.
By combination with above discussion, we have , which finishes the proof of Lemma 7.
Corollary 8. Suppose that is defined as above and , the value of reaches its maximum value at and .
Theorem 9. If is a connected graph with cut edges and vertices, then with equality if and only if (see Figure 3).
Proof. To determine the maximum Resistance-Harary index of the graph, we select such a connected graph so that it cut off all the cut edges for a complete graph by Lemma 3. Moreover, we can further choose by Lemmas 5 and 6, and let , . From the definition of the Resistance-Harary index and Lemma 4, we have By Lemma 7 and Corollary 8, we know that the equality holds if and only if , ; that is, . This completes the result.
Corollary 10. If is a tree with vertices, then , with equality if and only if .
Proof. Since is a tree with vertices, it has cut edges, and with equality if and only if by Theorem 9. This completes the result.
4. Conclusion
In this thesis, we considered the Resistance-Harary index of graphs with given number of cut edges and describe a graph with a maximum Resistance-Harary index. A problem raised naturally at this moment is, among all connected graphs with vertices and cut edges, which graph has the minimum Resistance-Harary index? We will continue to consider this problem in the nearest future.
Conflicts of Interest
The authors declare no competing financial interests.
Acknowledgments
Wang was supported by National Natural Science Foundation of China under Grant no. 11571135 and Humanities and Social Sciences of Ministry of Education Planning Fund under Grant no. 16YJA630032. Hua was supported in part by National Natural Science Foundation of China under Grant no. 11571135.