On the existence of non-golden signed graphs

Abstract

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