Connected Vertex-Edge Dominating Sets and Connected Vertex-Edge Domination Polynomials of Gem Graph Gn

Abstract

Let G be a simple connected graph of order n. Let D c ν e (G, i) be the family of connected vertex-edge dominating sets of G with cardinality i. The polynomial D c ν e (G, x) = D c υ e ( G x ) = ∑ ​ i = γ c υ e n ( G ) d c υ e ( G i ) x i is called the connected vertex-edge domination polynomial of G where d c ν e (G, i) is the number of vertex edge dominating sets of G. In this paper, we study some properties of connected vertex-edge domination polynomials of the Gem graph G n . Also we obtain some properties of D c ν e (G n , x) and their coefficients. Also, we find the recursive formula, to derive the connected vertex-edge dominating sets of the Gem graph G n .