Abstract
For any two distinct vertices and in a connected graph , let be the length of path and the D–distance between and of is defined as: , where the minimum is taken over all paths and the sum is taken over all vertices of path . The D-index of G is defined as . In this paper, we found a general formula that links the Wiener index with D-index of a regular graph G. Moreover, we obtained different formulas of many special irregular graphs.
1. Introduction
Throughout this paper, the nontrivial connected graphs were considered to be finite without multiple edges and loops [1]. As usual, which denotes the set of vertices of and represents the set of edges of , the order of is and the size of is . The vertex degree or is referred to the number of vertices incident to the vertex .
The standard distance between any two arbitrary vertices and of is the length of a shortest path joining to , and it is denoted by or . The diameter of a connected graph denoted by is the length of any longest geodesic [2]. Many researchers have determined different concepts of distances as well as ordinary distance, such as Steiner distance [3], width distance [4], superior distance [5], and signal distance [6]. All distances mentioned above depend on the path length in a connected graph .
In the present work, the concept of -distance between distinct vertices of was considered, in which it depends on the length of a path as well as the degrees of the vertices that lie on [7].
“The Wiener index of a graph is represented by and defined as the sum of distances between all pairs of vertices in a simple graph ”:
Based on the Wiener index, Hosoya introduced the Wiener polynomial (now called Hosoya polynomial) in 1988 [8]. The Wiener polynomial is defined as
There are a lot of papers on finding general formulas for Wiener polynomials for different types of distances (see [9, 10]).
Definition 1 (see [7]). “The D-distance between two vertices and of a connected graph is defined as
, where is the length of path and the minimum is taken over all paths in ”.
The total -distance or -index of a connected graph is defined as
Definition 2 (see [1]). It can be said that the graph is r-regular if every vertex in has r degree, that is, , for all , otherwise the graph is irregular.
Definition 3 (see [1]). A friendship graph is a graph of odd order , , which consists of triangles with a common vertex.
Let and be nontrivial disjoint connected graphs, then the Cartesian product graph has vertex set and edge set:and the strong product graph has vertex set and edge set:There are several papers about the -distance and the average of this distance, see References [11, 12, 13, 14, 15].
In this work, a relationship between the Wiener index and D-index of any regular graph was found. Furthermore, the D-index of some special graphs was obtained.
2. Preliminaries
In this section, we will mention some of the results that were found previously and others that will be established.
Theorem 1 (see [(16]). (1)(2)(3)(4)(5)
Theorem 2 (see [16, 17]). If and are nontrivial disjoint connected graphs, then(1)(2)If and in which and . Then, , where , , , and .
Lemma 1. For , we have(1)(2)
Proof. (1)From Theorem 2 (18), if and , , then Taking the derivative with respect to x of both sides and then putting , we get From Theorem 1 (20), we get(2)From Theorem 2 (19), if and , , then Taking the derivative with respect to x of both sides and then putting , we get From Theorem 1 (19), we get
3. The Index of D-Distance for a Connected r-Regular Graph
Theorem 3. For any connected regular graph of order , we have
Proof. Since is regular, then , , and every shortest path between and contains vertices with every vertex of degree .
Example 1. Let be a Petersen graph of ten orders, as shown in Figure 1.
It is clear that , then .
Corollary 1. (1)(2)(3)(4)
Proof. From Theorems 1 and 3, we get results listed in this corollary.
Corollary 2. For , we have(1)(2)
Proof. Since from and are all 3-regular and 5-regular, respectively, shown Figure 2, then from Lemma 1 and Theorem 3, we get the formulas given in Corollary 2.
(a)
(b)
4. The Index of D-Distance for Some Special Graph
n this section, we find the D-index of some special irregular graphs, such as the path graph , , the star graph , , the friendship graph , of order odd , , the wheel graph , , and the bipartite complete graph , , .
Theorem 4. (1)(2)Proof(1)Let be a path graph of order . Since , then . Also, by symmetric, , for all , and . Since , , then , for all . Hence,(2)Let be a star graph with order and let be the center of . It is obvious that Now, since the distance for each pair of end vertices is equal two, then .Hence, .
Theorem 5. (1)(2) where (3)
Proof. (1)Let , where and , for all . Then, there are three cases: Case 1: if , then , for all . Case 2: if ,, then , and the number of pairs of , when , for all , is . Case 3: if , , then the path consists from , Hence, . The number of pairs of when , , is . From the three previous cases, we get(2)Let , where and , for all . It is clear that Now, let , for all , then there are two cases: Case 1: if , then where the number of such pairs is . Case 2: if and is an even, then Also, the number of such pairs is . But when is odd, then and the number of such pairs is . Hence, from (18)–(21), we have the formula in Theorem 5 part 2.(3)Let , where and .The following cases are true in this graph: Case 1: if , then , this is true for all pairs of vertices. Case 2: if , then , and also this is true for all pairs of vertices. Case 3: if and If , then and there are pairs of vertices in this case that is true.From three cases, we conclude that
5. Conclusion
In this paper, we find a general formula giving a relation between the Wiener index and D-index for any regular graph, also we find the D-index for some special graphs.
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 called for a lack of conflicts of interest to publish this paper.
Acknowledgments
This paper was supported by the College Computer Sciences and Mathematics, University of Mosul, Republic of Iraq.