Abstract
The detour index of a connected graph is defined as the sum of the detour distances (lengths of longest paths) between unordered pairs of vertices of the graph. The detour index is used in various quantitative structure-property relationship and quantitative structure-activity relationship studies. In this paper, we characterize the minimum detour index among all tricyclic graphs, which attain the bounds.
1. Introduction
Let G be a simple and connected graph with and and be the neighbor vertex set of vertex u, then is called the degree of u. If , then G is called a c-cyclic graph. If and 3, then G is a tree, unicyclic graph, bicyclic graph, and tricyclic graph, respectively. Denote by the set of all tricyclic graphs of order n.
Let , where graphs for are defined in Figure 1. By [1, 2], we know that for any , can be obtained from by attaching trees to some of its vertices. We call the base of .
A block of the graph G is a maximal 2-connected subgraph of G. A cactus is a connected graph in which no edge lies in more than one cycle, such that each block of a cactus is either an edge or a cycle. A vertex shared by two or more cycles is called a cut vertex. In this paper, denote be the set of all cacti of order n and l cycles, where . The length of the cycles may be different and the length of each cycle is at least 3.
The concept of “topological index” was first proposed by Haruo Hosoya for characterizing the topological nature of a graph. Such graph invariants are usually related to the distance function .
The detour distance [3, 4] (also known under the name elongation) between vertices u and in G is the length of a longest path between them, denoted by . Note that for any ; see [5] for a discussion. The detour index of the graph G is defined as [4–9]
For a connected graph G with , let then
If we use the notion of the detour matrix [4], which is an matrix whose -element is with , then the detour index is equal to the half-sum of the (off diagonal) elements of the detour matrix. The detour index has been applied to chemistry, especially in quantitative structure-activity relationship (QSAR) studies; see [7, 10] for more details. A new branch cheminformatics is a combination of mathematics and chemistry. This branch studies QSAR/QSPR study, physicochemical properties and topological indices such as Zagreb Indices [11], Kirchhoff index [12], Hosoya index [13] and so on to predict physicochemical properties and biological activities of the chemical compounds theoretically.
In this paper, we consider the minimum detour index among all tricyclic graphs.
2. Preliminaries
In this section, we will introduce some useful lemmas and graph transformations.
2.1. Edge-Lifting Transformation [14, 15]
Let and be two graphs with and vertices, respectively. If G is the graph obtained from and by adding an edge between a vertex of and a vertex of , is the graph obtained by identifying of to of and adding a pendant edge to , then is called the edge-lifting transformation of G (see Figure 2).
Lemma 1 (see [16]). Let G be defined as in Figure 2, and is obtained from G by the edge-lifting transformation (see Figure 2). Then, .
Denote (see Figure 1).
By Lemma 1, we can verify that if attains the minimum detour index of all graphs in , then the following two conditions hold:(i)The base of is one of (ii)The graph is obtained from by attaching some pendant edges
Remark 1. In order to determine the tricyclic graphs which attain the minimum detour index of all graphs in , we just need to discuss the tricyclic graphs in , where .
2.2. Cycle-Edge Transformation
Let be a cactus as shown in Figure 3, where is the biggest cycle of , . Denote the vertex set . is the graph obtained from by deleting the edges and to , meanwhile adding the edges and to .
We say that is obtained from by the cycle-edge transformation (see Figure 3).
Lemma 2 (see [16]). Let be a cactus as shown in Figure 3 with , and be the cycle-edge transformation of (see Figure 3). Then, .
2.3. Cycle-Lifting Transformation
Let be a cactus as shown in Figure 4. Denote for . Let be the graph obtained from by deleting the edges for and adding the edges for .
We say that is the cycle-lifting transformation of (see Figure 4).
Lemma 3 (see [16]). Let be the cycle-lifting transformation of (see Figure 4). Then, .
2.4. Operation I
We define Operation I as follows. Let G and be a simple and connected graph as shown in Figure 5. be the path in a cycle. Denote , and be the graph obtained from G by deleting the edges , for and adding the edges for (see Figure 5).
Lemma 4. Let G and be the graph shown in Figure 5. Then, .
Proof. Let , and . For the vertices , obviouslyLet be the set of the longest path between and in G and be the set of the longest path between and in G, where .
Case 1. and , where .
Obviously, for , On the other hand, if and , then , where be the any one longest path between and in G. Therefore, , andCase 2. and .
Obviously, for , we haveCase 3. and .
Obviously, for , we haveCase 4. and .
Obviously, , andBy (5)–(9), we haveBy (3), (4), and (10), we have .
2.5. Operation II
We define Operation II as follows. Let G and be a simple and connected graph as shown in Figure 6. Denote be a cycle with length 3, , and be the graph obtained from G by deleting the edges for and adding the edges for (see Figure 6).
Lemma 5. Let G and be the graph shown in Figure 6. Then, , and the equality holds if and only if .
Proof. Let . For the vertices , , we haveFor the vertices , , we haveespecially, if , thenBy (11)–(16), we have and the equality holds if and only if .
Denote ; see Figure 7.
By Lemma 2–5, we can verify that if attains the minimum detour index of all graphs in , then .
Remark 2. In order to determine the tricyclic graphs which attain the minimum detour index of all graphs in , we just need to discuss the tricyclic graphs in , where ; see Figure 7.
2.6. Operation III
We define Operation III as follows. Let as shown in Figure 7. Denote and be the graph obtained from G by deleting the edges for and adding the edges for (see Figures 8–11).
Lemma 6. Let and be the graph in Figures 8–11. Then, and the equality holds if and only if .
Proof. Let . For the vertices ; , we haveBy (17)–(23), we have .
Similarly, we have and the equality holds if and only if (i = 2, 3, 4).
2.7. Operation IV
We define Operation IV as follows. Let as shown in Figure 12. Denote , and be the graph obtained from G by deleting the edges for and adding the edges for (see Figure 12).
Lemma 7. Let G and be the graph shown in Figure 12. Then, with equality holding if and only if .
Proof. Let , , we haveObviously,Therefore,and the equality holds if and only if .
Denote (see Figure 13).
By Lemma 6-7, we can verify that if attains the minimum detour index of all graphs in , then .
Remark 3. In order to determine the tricyclic graphs which attain the minimum detour index of all graphs in , we just need to discuss the tricyclic graphs in , where (see Figure 13).
3. Results and Discussion
From the discussions of Section 2, we can verify that if attains the minimum detour index of all graphs in , then where .
Theorem 1. Let be defined as in Figure 13.(1)When or , is the unique graph which attains the minimum detour index of all graphs in and .(2)When or , is the unique graph which attains the minimum detour index of all graphs in and .(3)When , and are the graphs which attain the minimum detour index of all graphs in and .
Proof. It can be checked directly thatTherefore,Similarly, we have(1)When or , obviously, is the unique graph which attains the minimum detour index of all graphs in and .(2)When or , obviously, is the unique graph which attains the minimum detour index of all graphs in and .(3)When , obviously, and are the graphs which attain the minimum detour index of all graphs in and .
4. Conclusions
Mathematical chemistry is an area of research in chemistry in which mathematical tools are used to solve problems of chemistry. Chemical graph theory is an important area of research in mathematical chemistry which deals with topology of molecular structures such as the mathematical study of isomerism and the development of topological descriptors or indices. In this paper, we first introduce some useful graph transformations, and then we determine the minimum detour index of all tricyclic graphs. In addition, all the corresponding extremal graphs are characterized.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest regarding the content and implications of this manuscript.
Acknowledgments
This work was supported by the open project of Anhui University (no. KF2019A01), the Natural Science Research Foundation of the Department of Education of Anhui Province (no. KJ2019A0817), and the Project of Teaching Team of Chizhou University (no. 2016XJXTD02).