On Characterization of Balance and Consistency Preserving d-Antipodal Signed Graphs

Abstract

A signed graph is an ordered pair Σ=(G,σ), where G is a graph and σ:E(G)⟶{+1,−1} is a mapping. For e∈E(G), σ(e) is called the sign of e and for any sub-graph H of G, σ(H)=∏e∈E(H)σ(e) is called the sign of H. A signed graph having a sign of each cycle +1 is called balanced. Two vertices in a graph G are called antipodal if dG(u,v)=diam(G). The antipodal graph A(G) of a graph G is the graph with a vertex set that is the same as that of G, and two vertices u,v in A(G) are adjacent if u,v are antipodal. By the d-antipodal graph GdA of a graph G, we refer to the union of G and A(G). Given a signed graph Σ=(G,σ), the signed graph ΣdA=(GdA,σd) is called the d-antipodal signed graph of G, where σd is defined as follows: σd(e)=σ(e)ife∈E(G)andotherwise,σd(e)=∏P∈Peσ(P), where Pe is the collection of all diametric paths in Σ connecting the end vertices of an antipodal edge e in ΣdA. In this article, the balance property and canonical consistency of d-antipodal signed graphs of Smith signed graphs (connected graphs having a highest eigenvalue of 2) are studied.