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

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 d G ( u , v ) = d i a m ( 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 G d A of a graph G , we refer to the union of G and A ( G ) . Given a signed graph Σ = ( G , σ ) , the signed graph Σ d A = ( G d A , σ d ) is called the d -antipodal signed graph of G , where σ d is defined as follows: σ d ( e ) = σ ( e ) if e ∈ E ( G ) and otherwise , σ d ( e ) = ∏ P ∈ P e σ ( P ) , where P e is the collection of all diametric paths in Σ connecting the end vertices of an antipodal edge e in Σ d A . 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.