Products of Distance Degree Regular and Distance Degree Injective Graphs

Abstract

The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex in G. The distance degree sequence (dds) of a vertex v in a graph G= (V,E) is a list of the number of vertices at distance 1, 2,…..,e(u) in that order, where e(u) denotes the eccentricity of u in G. Thus the sequence j di,di,dif,di,f 0 1 2 ^ h is the dds of the vertex vi in G where j di denotes number of vertices at distance j from vi . A graph is distance degree regular (DDR) graph if all vertices have the same dds. A graph is distance degree injective (DDI) graph if no two vertices have same dds. In this paper we consider Cartesian and normal products of DDR and DDI graphs. Some structural results have been obtained along with some characterizations.