VERTEX-TO-EDGE CENTERS W.R.T. D-DISTANCE

Abstract

Between any two vertices of a graph we can define many distances, the usual distance, detour distance, superior distance, signal distance, degree distance etc.  In some of these distances only the lengths of various paths were considered.  By considering the degrees of various vertices present in a path, in addition to the length of the path, in an earlier article we introduced the concept of D-distance, d^D(u, v), in graphs.  In this article we study D-distance between a vertex and an edge of a graph and determine the eccentrics, radius and diameters of some classes of graphs.  We also obtain relations between their eccentrics and also determine centers of graphs.  Further, we prove that for any graph G either C^D(G) ⊆ C\(_1^{D}\)(G) or C\(_1^{D}\)(G) ⊆ C^D(G).