On the Existence of Non-golden Signed Graphs

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 eigenvector 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 Sx(Γ,C) = { |xrxs| | rs ∈ E(C) }. In this paper we show that non-golden bicyclic graphs with frustration index 1 exist for each n ≥ 5.