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 ${ u_ i ,\, v_ i \,|\, 0 \leq i \leq n - 1 }$ and edge set ${ u_ i u_ {i + 1} , u_ i v i , v i v_{ i + k}\, |\, 0\leq i \leq n - 1, \hbox{ subscripts reduced modulo } n }$. 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.