On Numerical Invariant of Graph

Abstract Let G = (V (G), E(G)) be a finite graph with n vertices, where V (G) = {v i, … , v n} denote vertex set of G and E(G) denote an edge set G. In this paper, we have introduced the graph G p , q for G. The motivation for introducing the graph G p , q are as follows: Many mathematicians studied the properties of graph G by using boundary operator and co-boundary operator which consider only the relation between vertices and an edges of a graph G. There is no place for complete subgraph (whose vertices greater than 2) of a graph G in boundary operator and co-boundary operator of G. Hence to study the relation between complete subgraph of G and also to study the properties of G, we have introduced an edge-boundary operator and an edge-co-boundary operator on G p,q .