On the Independent Neighborhood Polynomial of the Cartesian Product of Some Special Graphs

Abstract

Two vertices x, y of a graph G are adjacent, or neighbors, if xy is an edge of G. A set S of vertices in a graph G is a neighborhood set if G =[v∈ShN[v]i where hN[v]i is the subgraph induced by v and all the vertices adjacent to v. If no two of the elements of S are adjacent, then S is called an independent neighborhood set. The independent neighborhood polynomial of G of order m is Ni(G, x) = Xm j=ηi(G) ni(G, j)xj where ni(G, j) is the number of independent neighborhood set of G of size j and ηi(G) is the minimum cardinality of an independent neighborhood set of G. This paper investigates the independent neighborhood polynomial of the Cartesian product of some special graphs.