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

A signed graph is an ordered pair s=(G,s), where G is a graph and s:E(G)?{+1,-1} is a mapping. For e & ISIN;E(G), s(e) is called the sign of e and for any sub-graph H of G, s(H),E(H)s(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 s=(G,s), the signed graph sdA=(GdA,sd) is called the d-antipodal signed graph of G, where sd is defined as follows: sd(e)=s(e)if e ? E(G)andotherwise,sd(e)=?Pes(P), where Pe is the collection of all diametric paths in s connecting the end vertices of an antipodal edge e in sdA. 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.