Balanced graphs from lexicographic products of open neighborhood graphs

Abstract

This paper investigates the structural balance properties of Lexicographic Product graphs formed by combining standard graph classes with their corresponding Open Neighborhood graphs. Specifically, we construct the Lexicographic Product $G' = G[N(G)]$, where $G$ is a standard graph and $N(G)$ denotes its open neighborhood graph. We examine the balance of the resulting signed graphs under various edge sign assignments. For each graph class considered, we rigorously demonstrate that the resulting Lexicographic Product graph is both regular and structurally balanced. The sign assignment methodology is derived from adjacency relationships in both $G$ and $N(G)$. Through detailed examples and structural proofs, we confirm that the signed Lexicographic Product graphs consistently exhibit balance, underscoring their significance in the study of signed and structured networks.