Geodesic Graphs

Abstract

In this paper, we introduce the concept of geodesic graph at a vertex of a connected graph and investigate its properties. We determine the bounds for the number of edges of the geodesic graph. We prove that an edge of a graph is a cut edge if and only if it is a cut edge of each of its geodesic graphs. Also, we characterize a bipartite graph as well as a geodetic graph in terms of its geodesic graph.