Edge perturbation on signed graphs with clusters: Adjacency and Laplacian eigenvalues

Abstract

Let Γ=(G,σΓ) be a signed graph, where G is the underlying simple graph and σΓ:E(G)⟶{+1,−1} is the sign function on the edges of G. A (c,s)-cluster in Γ is a pair of vertex subsets (C,S), where C is a set of cardinality |C|=c≥2 of pairwise co-neighbour vertices sharing the same set S of s neighbours. For each signed graph Λ of order c we consider the graph Γ(Λ) obtained by adding the edges of Λ, after suitably identifying C and V(Λ). It turns out that Γ(Λ) and Γ(Λ′) share part of their adjacency (resp. Laplacian) spectrum if Λ and Λ′ both show the sign-based regularity known as net-regularity (resp. negative regularity). Our results offer a generalization to signed graph of some theorems by Domingos Cardoso and Oscar Rojo concerning edge perturbations on (unsigned) graphs with clusters.