Equi Neighbor Polynomial of Some Binary Graph Operations

Abstract

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 \(u \in G,\) let \(N(u)\) be the set of all vertices of \(G\) which are adjacent to \(u\) in \(G.\) Then for \(0\leq i \leq n-1\), the \(i\)-equi neighbor set of \(G\) is defined as: \(N_{e}(G,i)=\{(u,v):u, v\in V, u\neq v\) and \(|N(u)|=|N(v)|=i\}.\) The equi-neighbor polynomial \(N_{e}[G;x]\) of \(G\) is defined as \(N_{e}[G;x]=\sum_{i=0}^{(n-1)} |N_{e}(G,i)| x^{i}.\) In this paper we discuss the equi-neighbor polynomial of graphs obtained by some binary graph operations.