Unbalanced signed graphs with extremal spectral radius or index

Abstract

Let G˙ = (G, σ) be a signed graph, and let ρ(G˙ ) (resp. λ1(G˙ ) ) denote the spectral radius (resp. the index) of the adjacency matrix AG˙. In this paper we detect the signed graphs achieving the minimum spectral radius m(SRn) , the maximum spectral radius M(SRn) , the minimum index m(In) and the maximum index M(In) in the set Un of all unbalanced connected signed graphs with n⩾ 3 vertices. From the explicit computation of the four extremal values it turns out that the difference m(SRn) - m(In) for n⩾ 8 strictly increases with n and tends to 1, whereas M(SRn) - M(In) strictly decreases and tends to 0.