kk-Combinations of an unlabelled graph

In this paper we extend the notion of the binomial coefficient (nk)(nk) into a new notion ([G]k)([G]k), where [G][G] is an unlabelled graph with nn vertices and 0⩽k⩽n0⩽k⩽n. We call ([G]k)([G]k) as the graph binomial coefficient and a version of the graph binomial expansion is also studied. As an application of this notion, we enumerate the number of ways to color vertices of a path and beads of a necklace.