Distance degree graphs in the Cartesian product of graphs

Abstract

AbstractThe distance degree sequences of graphs were studied for the purpose of distinguish- ing chemical isomers by their graph structure. Degree sequence distribution provides a mechanism for classifying graphs according to their topology. In [11] Medha et al. listed two open problems (1) Characterize DDR graphs of diameter higher than 3 and (2) Does there exist DDI r-regular graph for r>4? In this paper, we settle this problem for the Cartesian product of graphs. We show that a graph G ≅ H1H2 is DDR if and only if both H1 and H2 are DDR. Also, it is interesting to see that there does not exist a DDI graphs, SI graphs, geodetic graphs and UDD graphs in the Cartesian product. Even though the factors have these properties, there does not exist such graphs in the Cartesian product.Keywords: distance degree regular graphsdistance degree injective graphsstatus injective graphsgeodetic graphsuniform distance distribution graphsCartesian productAMS Classification: 05C3805C76