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Title: Applications of Estrada Indices and Energy to a family of compound graphs
Author: Andrade, Enide
Pizarro, Pamela
Robbiano, Maria
San Martin, B.
Tapia, Katherine
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.
Peer review: yes
DOI: 10.1016/j.laa.2017.06.035
ISSN: 0024-3795
Appears in Collections:CIDMA - Artigos
OGTCG - Artigos

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