Unbalanced signed graphs with extremal spectral radius or index

Abstract

Let dot{G}=(G,sigma ) be a signed graph, and let rho (dot{G}) (resp. lambda _1(dot{G})) denote the spectral radius (resp. the index) of the adjacency matrix A_{dot{G}}. In this paper we detect the signed graphs achieving the minimum spectral radius m(mathcal S mathcal R_n), the maximum spectral radius M(mathcal S mathcal R_n), the minimum index m(mathcal I_n) and the maximum index M(mathcal I_n) in the set mathcal U_n of all unbalanced connected signed graphs with ngeqslant 3 vertices. From the explicit computation of the four extremal values it turns out that the difference m(mathcal S mathcal R_n)-m(mathcal I_n) for n geqslant 8 strictly increases with n and tends to 1, whereas M(mathcal S mathcal R_n)- M(mathcal I_n) strictly decreases and tends to 0.