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title:  Edge perturbation on graphs with clusters: Adjacency, Laplacian and signless Laplacian eigenvalues 
authors:  Cardoso, Domingos M. Rojo, Oscar 
keywords:  Adjacency, Laplacian and signless Laplacian spectra of graphs Graph cluste Algebraic connectivity Laplacian index Adjacency index 
issue date:  Jan2017 
publisher:  Elsevier 
abstract:  Let G be a simple undirected graph of order n. A cluster in G of order c and degree s, is a pair of vertex subsets (C, S), where C is a set of cardinality C =c ≥2 of pairwise coneighbor vertices sharing the same set S of s neighbors. Assuming that the graph G has k≥1 clusters (C_1, S_1), ..., (C_k, S_k), consider a family of k graphs H_1, ..., H_k and the graph G(H_1, ..., H_k) which is obtained from G after adding the edges of the graphs H_1, ..., H_ k whose vertex set of each H_j is identified with C_j, for j=1, ..., k. The Laplacian eigenvalues of G(H_1, ..., H_k)remain the same, independently of the graphs H_1, ..., H_k, with the exception of C_1 +···+C_k −k of them. These new Laplacian eigenvalues are determined using a unified approach which can also be applied to the determination of a same number of adjacency and signless Laplacian eigenvalues when the graphs H_1, ..., H_k are regular. The Faria’s lower bound on the multiplicity of the Laplacian eigenvalue 1 of a graph with pendant vertices is generalized. Furthermore, the algebraic connectivity and the Laplacian index of G(H_1, ..., H_k) remain the same, independently of the graphs H_1, ..., H_k. 
URI:  http://hdl.handle.net/10773/16521 
ISSN:  00243795 
publisher version/DOI:  http://dx.doi.org/10.1016/j.laa.2016.09.031 
source:  Linear Algebra and its Applications 
appears in collections  CIDMA  Artigos

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