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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 .

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