ON THE EXISTENCE OF NON-GOLDEN SIGNED GRAPHS

A signed graph is a pair Gamma = (G, sigma), where G = (V (G), E(G)) is a graph and sigma : E (G) -> {+1, -1} is the sign function on the edges of G. For a signed graph we consider the least eigenvalue lambda (Gamma) of the Laplacian matrix defined as L(Gamma) = D(G) -A (Gamma), where D(G) is the matrix of vertices degrees of G and A(Gamma) is the signed adjacency matrix. An unbalanced signed bicyclic graph is said to be golden if it is switching equivalent to a graph Gamma satisfying the following property: there exists a cycle C in Gamma and a lambda(Gamma)-eigenvector x such that the unique negative edge pq of Gamma belongs to C and detects the minimum of the set S-x(Gamma,C) = { vertical bar x(r)x(s)vertical bar vertical bar rs is an element of E(C) }. In this paper we show that non-golden bicyclic graphs with frustration index 1 exist for each n >= 5.