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 \(Nₑ[G;x]=∑ᵢ₌₀⁽ⁿ⁻¹⁾ |Nₑ(G,i)| xⁱ.\) In this paper we discuss the equi-neighbor polynomial of graphs obtained by some binary graph operations.