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: \ (N₄ (G, i) =\ (u, v): u, v V, u v\) and \ (|N (u) |=|N (v) |=i\. \) The equi-neighbor polynomial \ (N₄G;x\) of \ (G\) is defined as \ (N₄G;x=₈=₀^ (n-1) |N₄ (G, i) | x^i. \) In this paper we discuss the equi-neighbor polynomial of graphs obtained by some binary graph operations.