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: Andrade, EnidePizarro, PamelaRobbiano, MariaSan Martin, B.Tapia, Katherine keywords: Compound graphEstrada indexLaplacian Estrada indexSignless Laplacian Estrada indexHypoenergetic graphIsospectral 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 collections CIDMA - Artigos

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