Some bounds for the largest eigenvalue of a signed graph

Abstract

For a vertex i of a signed graph, let di, mi and T-i denote its degree, average 2-degree and the number of unbalanced triangles passing through i, respectively. We prove that (Equation presented) where ρ stands for the largest eigenvalue. This bound is tight at least for regular signed graphs that are switching equivalent to their underlying graphs. We also derive a general lower bound for ρ and certain practical consequences. A discussion, including the cases of equality in inequalities obtained and some examples, is given.