Polynomial Representation and Degree Sequence of Graphs Resulting from Some Graph Operations (European Journal of Pure and Applied Mathematics, 2024)
Let $G = (V(G),E(G))$ be a graph with degree sequence $\langle d_1,d_2, …, d_n \rangle$, where $d_1 \ge d_2 \ge … \ge d_n$. The polynomial representation of $G$ is given by $f_G(x) = \displaystyle \sum_{i=1}^n x^{d_i} = \sum_{k=1}^{Δ(G)}a_kx^{k}$, where $a_k$ 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.
Keywords: polynomial representation; line graph; edge corona