Edge perturbation on graphs with clusters: Adjacency, Laplacian and signless Laplacian eigenvalues - Cardoso, Domingos M. (2017)
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 co-neighbor 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.