Laplacian spread of graphs: lower bounds and relations with invariant parameters

• Autor(a) principal: Andrade, Enide
• Data de Publicação: 2015
• Outros Autores: Cardoso, Domingos, Robbiano, Maria, Rodriguez, Jonnathan
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10773/15030
• topic: Spectral Graph Theory; Matrix spread; Laplacian Spread

The spread of an $n\times n$ complex matrix $B$ with eigenvalues $\beta _{1},\beta _{2},\ldots ,\beta _{n}$ is defined by \begin{equation*} s\left( B\right) =\max_{i,j}\left\vert \beta _{i}-\beta _{j}\right\vert , \end{equation*}% where the maximum is taken over all pairs of eigenvalues of $B$. Let $G$ be a graph on $n$ vertices. The concept of Laplacian spread of $G$ is defined by the difference between the largest and the second smallest Laplacian eigenvalue of $G$. In this work, by combining old techniques of interlacing eigenvalues and rank $1$ perturbation matrices new lower bounds on the Laplacian spread of graphs are deduced, some of them involving invariant parameters of graphs, as it is the case of the bandwidth, independence number and vertex connectivity.