### 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***(Γ,**_{x}*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.#### Keywords

Signed Graph; Bicyclic Graph; Laplacian; Least Eigenvalue; Theta-Graph

#### Full Text:

PDFDOI: http://dx.doi.org/10.1478/AAPP.96S2A2

Copyright (c) 2018 Maurizio Brunetti

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