Equi-neighbor Polynomial Equivalent Classes of Graphs

Let G(V,E) be a simple graph of order n with vertex set V and edge set E. Let (u,v) denotes 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] = \(Σ^{(n−1)}_{i=0}\) |Ne(G, i)|xi. Two graphs G and H are said to be ENP− equivalent if and only if Ne[G; x]=Ne[H; x]. A graph H is said to be ENP-unique if H is ENP-equivalent to itself. A graph G is said to be ENP-cardinal unique if there does not exist a graph H with the same number of vertices as G such that H is equivalent to G. This paper identifies several ENP-unique graphs, explores ENP-equivalent graph classes, and characterizes specific ENP-ardinal unique graph classes.