  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.
