ON TOPOLOGIES INDUCED BY HOP NEIGHBORHOODS OF GRAPHS

Abstract

In this paper, we investigate structural properties of simple connected undirected graphs through a topology on the vertex set induced by their hop neighborhoods. In this construction, we characterize the classes of graphs that induce the discrete and indiscrete topologies. We also characterize the singleton sets in the topology induced by the join of a complete graph and an arbitrary graph, $K_n + G$; thus, the corresponding topologies induced by fan and wheel graphs are determined. Finally, specific topologies induced by the path and the cycle are obtained.