Unbalanced signed graphs with extremal spectral radius or index

Abstract

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