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\section{On the spectra of some graphs like weighted rooted trees} By {\sl Ros\'{a}rio, Helena Gomes and Enide Andrade Martins}. 

\medskip  \noindent Let $G$ be a weighted rooted graph of $k$ levels such that, for $j\in\{2,\dots ,k\}$  \begin{enumerate} \item each vertex at level $j$ is adjacent to one vertex at level $j-1$ and all edges joining a vertex at level $j$ with a vertex at level $j-1$ have the same weight, where the weight is a positive real number.  \item if two vertices at level $j$ are adjacent then they are adjacent to the same vertex at level $j-1$ and all edges joining two vertices at level $j $ have the same weight.  \item two vertices at level $j$ have the same degree.  \item there is not a vertex at level $j$ adjacent to others two vertices at the same level. \end{enumerate}  In this talk we give a complete characterization of the eigenvalues of the Laplacian matrix of $G$ (analogous characterization can be done for the adjacency matrix of $G)$). By application of the these results, we derive an upper bound on the largest eigenvalue of a graph defined by a weighted tree and a weigthed triangle attached, by one of its vertices, to a pendant vertex of the tree.
%Graph; Laplacian matrix; Adjacency matrix; Eigenvalues","05C50","","","10:24:56","Sat Mar 29 2008","89.214.211.4"
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