Repositório Institucional da Universidade de Aveiro > CIDMA - Centro de Investigação e Desenvolvimento em Matemática e Aplicações > CIDMA - Artigos >
 Applications of Estrada Indices and Energy to a family of compound graphs
Please use this identifier to cite or link to this item http://hdl.handle.net/10773/18245

title: Applications of Estrada Indices and Energy to a family of compound graphs
authors: Enide Andrade
Pamela Pizarro
Maria Robbiano
B. San Martin
Katherine Tapia
keywords: Compound graph
Estrada index
Laplacian Estrada index
Signless Laplacian Estrada index
Hypoenergetic graph
Isospectral graph
issue date: Nov-2017
publisher: Elsevier
abstract: To track the gradual change of the adjacency matrix of a simple graph $\mathcal{G}$ into the signless Laplacian matrix, V. Nikiforov in \cite{NKF} suggested the study of the convex linear combination $A_{\alpha }$ (\textit{$\alpha$-adjacency matrix}), \[A_{\alpha }\left( \mathcal{G}\right)=\alpha D\left( \mathcal{G}\right) +\left( 1-\alpha \right) A\left( \mathcal{G}\right),\] for $\alpha \in \left[ 0,1\right]$, where $A\left( \mathcal{G}\right)$ and $D\left( \mathcal{G}\right)$ are the adjacency and the diagonal vertex degrees matrices of $\mathcal{G}$, respectively. Taking this definition as an idea the next matrix was considered for $a,b \in \mathbb{R}$. The matrix $A_{a,b}$ defined by $$ A_{a,b}\left( \mathcal{G}\right) =a D\left( \mathcal{G}\right) + b A\left(\mathcal{G}\right),$$ extends the previous $\alpha$-adjacency matrix. This matrix is designated the \textit{$(a,b)$-adjacency matrix of $\mathcal{G}$}. Both adjacency matrices are examples of universal matrices already studied by W. Haemers. In this paper, we study the $(a,b)$-adjacency spectra for a family of compound graphs formed by disjoint balanced trees whose roots are identified to the vertices of a given graph. In consequence, new families of cospectral (adjacency, Laplacian and signless Laplacian) graphs, new hypoenergetic graphs (graphs whose energy is less than its vertex number) and new explicit formulae for Estrada, signless Laplacian Estrada and Laplacian Estrada indices of graphs were obtained. Moreover, sharp upper bounds of the above indices for caterpillars, in terms of length of the path and of the maximum number of its pendant vertices, are given.
URI: http://hdl.handle.net/10773/18245
ISSN: 0024-3795
publisher version/DOI: https://doi.org/10.1016/j.laa.2017.06.035
source: Linear Algebra and its Applications
appears in collectionsCIDMA - Artigos

files in this item

file description sizeformat
LAA_D_17Revised.pdfMain article341.26 kBAdobe PDFview/open
Restrict Access. You can Request a copy!

Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.


Valid XHTML 1.0! RCAAP OpenAIRE DeGóis
ria-repositorio@ua.pt - Copyright ©   Universidade de Aveiro - RIA Statistics - Powered by MIT's DSpace software, Version 1.6.2