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title:  Applications of Estrada Indices and Energy to a family of compound graphs 
authors:  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:  Nov2017 
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:  00243795 
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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