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.