Equi Neighbor Polynomial of Some Binary Graph Operations

Let G(V, E) be a simple graph of order n with vertex set V and edge set E. Let (u, v) denote an unordered vertex pair of distinct vertices of G. For a vertex u ∈ G, let N(u) be the set of all vertices of G which are adjacent to u in G. Then for 0 ≤ i ≤ n − 1, the i-equi neighbor set of G is defined as: Ne(G, i) = {(u, v): u, v ∈ V, u , v and |N(u)| = |N(v)| = i}. The equi-neighbor polynomial Ne[G; x] of G is defined as Ne[G; x] = P(i=n−01) |Ne(G, i)|xi. In this paper we discuss the equi-neighbor polynomial of graphs obtained by some binary graph operations. © 2024 the Author(s), licensee Combinatorial Press.