Polynomial Representation and Degree Sequence of Graphs Resulting from Some Graph Operations

Let $G = (V(G),E(G))$ be a graph with degree sequence d₁,d₂, ⋯, dₙ, where d₁ ≥ d₂ ≥ ⋯ ≥ dₙ. The polynomial representation of G is given by fG(x) = ∑ᵢ₌₁ⁿ xdᵢ = ∑ₖ₌₁Δ(G)aₖxᵏ, where aₖ is the number of vertices of G having degree k for each i = 1,2,⋯ n = Δ(G). In this paper, we give the polynomial representation of the complement and line graph of a graph, the shadow graph, complementary prism, edge corona, strong product, symmetric product, and disjunction of two graphs.