Unbalanced signed graphs with extremal spectral radius or index

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 A( ˙G) . 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 U_n 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.