Average circular D-distance and circular D-Wiener index of K-regular graphs

The circular -distance between nodes of a graph is obtained by the sum of detour -distance and -distance. The average circular -distance between the nodes of a graph is the sum of average of the detour -distance and -distances. In this paper, we deal with the average circular -distance between nodes of graph. We compute the relation between circular Wiener index and circular -Wiener index of -regular graphs. We obtained results on circular -Wiener index of some special graphs


Introduction
The idea of distance is one of the significant idea in investigation of graphs. It is used to test the isomorphism of graphs, connectivity problems and convexity of graphs etc.
In previous article (in [15]), the authors presented the idea of circular D -distance in graphs by adding detour D -distance and D -distance. A relation between Wiener index for k -regular graphs and D -index (was obtained by Ahmed Mohammed Ali and Asmaa Salah Aziz in [1]).
In this article we introduce, the concept of average circular D -distance between nodes of a graph and study some of its properties. Further we compute the average circular D -distance of some classes of graphs. We introduce the concept of Wiener detour D -index of a graph H . We obtain a relation between detour Wiener index and detour D -Wiener index of any regular graph H . Further we compute detour D -Wiener index of some special graphs. Finally we obtain a relation between circular Wiener index and circular D -Wiener index of regular graph H . Furthermore, this distance is also used in molecular dynamics of physics, crystallography, lattice statistics and physics.
Throughout this article we consider connected and simple graphs. For any unexplained terminology and symbols, we refer the book [2].

Average circular D-distance
In this section we given some definitions for later use.

Results on average circular D-distance
Now we prove some results on AVCDD between nodes. We begin with theorem which connects the number of nodes and AVCDD. This leads to some more results.

Results on some families of graphs
Here we compute the AVCDD for some families of graphs.
The AVCDD of a path graph n P is 4 n a n , where Proof: Consider the circular − D distance matrix of the path graph n P , which is  nn symmetric matrix.

The index of detour D-distance for connected k-regular graphs
Here we begin with some definitions on Wiener index and detour − D Wiener index.
The proof is similar to that of 6.5 below.

The detour D-index of some special graphs
Now we find the detour − D index of graphs which are not regular. Now we begin with path graph.

Circular D-Wiener index of k-regular graphs
Here we introduce the definition of circular $D$-Wiener index and compute the connection between circular D − Wiener index of k − regular graphs and circular Wiener index.

Conclusion
In this paper, we discuss properties of AVCDD and we find the average circular D − distance of some families of graphs. Further, we find a formula giving a connection between the detour Wiener index and detour D − Wiener index. We find the connection between circular Wiener index and circular D − Wiener index of k − regular graphs.
In future work, we find average circular D − distance between arcs of a graph and we can find the relation between circular D − Wiener index and average circular D − distance between arcs of a graph.