On the monophonic and monophonic domination polynomial of a graph

Abstract

A set S of vertices of a graph G is a monophonic set of G if each vertex u of G lies on an u ? v monophonic path in G for some u, v ? S. M ? V (G) is said to be a monophonic dominating set if it is both a monophonic set and a dominating set. Let M(G, i) be the family of monophonic sets of a graph G with cardinality i and let m(G, i) = |M(G, i)|. Then the monophonic polynomial M(G, x) of G is defined as M(G, x) = ??i=m(G) m(G, i)x?, where m(G) is the monophonic number of G. In this article, we have introduced monophonic domination polynomial of a graph. We have computed the monophonic and monophonic domination polynomials of some specific graphs. In addition, monophonic and monophonic domination polynomial of the Corona product of two graphs is derived.