Please cite this article as: F. Belardo, M. Brunetti and A. Ciampella, Edge perturbation on signed graphs with clusters: Adjacency and Laplacian eigenvalues, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2019.02.018. Discrete Applied Mathematics xxx (xxxx) xxx Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Edge perturbation on signed graphs with clusters: Adjacency and Laplacian eigenvalues Francesco Belardo, Maurizio Brunetti ∗ , Adriana Ciampella Dipartimento di Matematica e Applicazioni ‘‘R. Caccioppoli’’, Università di Napoli ‘‘Federico II’’, Naples, Italy article info Article history: Received 14 May 2018 Received in revised form 8 January 2019 Accepted 6 February 2019 Available online xxxx Dedicated to the 65th birthday of Domingos Cardoso Keywords: Signed graphs Cluster Laplacian Eigenvalue Spectrum abstract Let Γ = (G,σ Γ ) be a signed graph, where G is the underlying simple graph and σ Γ : 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. © 2019 Elsevier B.V. All rights reserved. 1. Introduction A signed graph Γ is a pair (G,σ Γ ), where G = (V (G), E (G)) is a simple graph (the underlying graph) and σ Γ : E (G) → {+1, −1} is a sign function on the edges of G (the signature). Signed graphs first appeared in 1950’s with fruitful applications in social psychology, and can be seen as a generalization of (unsigned) graphs, for which usual edges are taken positive. Most of the usual graph theory glossary directly extends to signed graphs. For instance, the order |Γ | of Γ is just the order of G, the degree deg(v) of a vertex v in Γ is simply its degree in G (independently of the signs of incident edges), and Γ is said to be k-regular if deg(v) = k for all v ∈ G. Whenever a subgraph G ′ of the underlying graph is considered, we assume defined on G ′ the restriction of the original sign function. A cycle C s of order s in Γ is said to be positive (resp. negative) if its sign, given by sign(C s ) = ∏ e∈E(Cs ) σ Γ (e), is equal to 1 (resp. −1). Therefore a cycle C s is positive if and only if the number of its negative edges is even. A signed graph is balanced if all cycles are positive; otherwise it is unbalanced [13]. If the signed graph Γ contains just positive (resp. negative) edges we denote it by (G, +) (resp. (G, −)), and call its signature all-positive (resp. all-negative). When dealing with signed graphs, the concept of (signature) switching cannot be avoided. For Γ = (G,σ Γ ) and U ⊂ V (G), let Γ U be the signed graph obtained from Γ by reversing the signature of the edges in the cut [U , V (G) \ U ], namely σ Γ U (e) =−σ Γ (e) for any edge e between U and V (G) \ U , and σ Γ U (e) = σ Γ (e) otherwise. The signed graph Γ U is said to be (signature) switching equivalent to Γ . In fact, switching equivalent signed graphs can be considered as (switching) isomorphic graphs and their signatures are said to be equivalent. Observe also that switching equivalent graphs have the same set of positive cycles. ∗ Corresponding author. E-mail addresses: fbelardo@unina.it (F. Belardo), mbrunett@unina.it (M. Brunetti), ciampell@unina.it (A. Ciampella). https://doi.org/10.1016/j.dam.2019.02.018 0166-218X/© 2019 Elsevier B.V. All rights reserved.