Exact Wiener Index of the Direct Product of a Path and a Wheel Graph

Shi-Jie Duan,Feng Li,Benedek Nagy

doi:10.1155/2022/1077549

Abstract

The direct product is one of the most important methods to construct large-scale graphs using existing small-scale graphs, and the topological structure parameters of the constructed large-scale graphs can be derived from small-scale graphs. For a simple undirected graph G , its Wiener index W G is defined as the sum of the distances between all different unordered pairs of vertices in the graph. Path is one of the most common and useful graphs, and it is found in almost all virtual and real networks; wheel graph is a kind of graph with good properties and convenient construction. In this paper, the exact value of the Wiener index of the direct product of a path and a wheel graph is given, and the obtained Wiener index is only derived from the orders of the two factor graphs.

Highlights

In this paper, a simple graph G refers to the set of vertices V(G) and the set of different unordered vertex pairs {a, b} in V(G), i.e., the edge set E(G) of G
We mainly research the Wiener index of the direct product of a path and a wheel graph, so we review the definitions of paths and wheel graphs
We summarize each term of formula (7), and by using the properties of the direct product, we get the final result of Case 2 as follows: m− 1 m− 1

Summary

Introduction

A simple graph G refers to the set of vertices V(G) and the set of different unordered vertex pairs {a, b} in V(G), i.e., the edge set E(G) of G. Two different vertices (ai, bj) and (ak, bl) of the direct product graph G × H are adjacent to each other, if and only if (ai, ak) ∈ E(G) and (bj, bl) ∈ E(H). E graph obtained by connecting two vertices a0 and an of a a0an-path with an edge is a cycle Cn+1 with order n + 1.

Results

Conclusion