## PATHOS DEGREE PRIME GRAPH OF A TREE

## PATHOS DEGREE PRIME GRAPH OF A TREE

*H. M. Nagesh*

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## 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 0 i and P 0 j (i 6= j) of a pathos of T have a vertex in common; or one corresponds to the path P 0 and the other to a vertex v and P 0 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 pla- narity; outerplanarity; maximal outerplanarity; minimally nonouterplanarity; Eulerian; and Hamiltonian properties these graphs.

## Keywords

Crossing number, inner vertex number, pathos, path number.