On the number of geodesics of Petersen graph $GP(n,2)$

Abstract

In any network, the interconnection of nodes by means of geodesics and the number of geodesics existing between nodes are important. There exists a class of centrality measures based on the number of geodesics passing through a vertex. Betweenness centrality indicates the betweenness of a vertex or how often a vertex appears on geodesics between other vertices. It has wide applications in the analysis of networks. Consider $GP(n,k)$. For each $n$ and $k(n>2k)$, the generalized Petersen graph $GP(n,k)$ is a trivalent graph with vertex set $\{u_i,v_i|0\le i\le n-1\}$ and edge set $\{u_i{u}_{i+1},u_i{v}_i,v_i{v}_{i+k}|0\le i\le n-1,subscripts\; reduced\; modulon\}$. There are three kinds of edges namely outer edges, spokes and inner edges. The outer vertices generate an $n$-cycle called outer cycle and inner vertices generate one or more inner cycles. In this paper, we consider $GP(n,2)$ and find expressions for the number of geodesics and betweenness centrality.

17 pages