Pathos degree prime graph of a tree

Abstract

Let T be a tree of order n (n ? 2). A pathos degree prime graph of T, written PDP(T), is a graph whose vertices are the vertices and paths of a pathos of T, with two vertices of PDP(T) adjacent whenever the degree of the corresponding vertices of T are unequal and relatively prime; or the corresponding paths P?? and P?? (i ? j) of a pathos of T have a vertex in common; or one corresponds to the path P? and the other to a vertex v and P? begins (or ends) at v such that v is a pendant vertex in T. We look at some properties of this graph operator. For this class of graphs we discuss the planarity; outerplanarity; maximal outerplanarity; minimally nonouterplanarity; Eulerian; and Hamiltonian properties these graphs.