Limit points for the spectral radii of unbalanced signed graphs

The spectral radius of a signed graph Γ = (G,σ) is the largest absolute value of its adjacency eigenvalues. In this paper we prove that sequences of unbalanced signed graphs suffice to retrieve all the possible limit points for the spectral radii of signed graphs. In order to achieve this result it turns out to be decisive a revisiting of the celebrated Shearer’s construction of the sequence of simple (unsigned) graphs whose index tends to a fixed real number a ⩾√︁(2 + √5). Additionally, our technique helps to detect infinite pairs of non-isomorphic cospectral pairs with unbalanced quadrangles.