# Bled'11 - 7th Slovenian International Conference on Graph Theory

19-25 June 2011
Bled, Slovenia
Europe/Ljubljana timezone
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# On a graph with the smallest eigenvalue at least $-\frac{1}{2}(3+\sqrt{5})$ --- an irreducibility of Hoffman graphs ---

Presented by Dr. Tetsuji TANIGUCHI
Type: Oral presentation
Track: Graph Spectra and its Applications

## Content

Hoffman graph may be regarded as a graph obtained by adding cliques to a simple graph, and is formally defined to be a graph which consists of the vertices of the simple graph and the fat vertices expressing the cliques. We may consider that the simple graphs are Hoffman graphs without fat vertices. In \cite{hlg}, R.~Woo and A.~Neumaier considered the sum" of Hoffman graphs, but they did not formulate the concept of the sum" since they treated it secondarily in order to consider another concept. By a process to study \cite{T} and \cite{T2}, we discovered the concepts of the sum" and the irreducible decomposition" of Hoffman graphs. In this talk, we deal with an irreducibility of Hoffman graphs, and, in particular, Hoffman graphs with the smallest eigenvalue at least $-\frac{1}{2}(3+\sqrt{5})$. \begin{thebibliography}{3} \bibitem{hlg} R.~Woo and A.~Neumaier, On graphs whose smallest eigenvalue is at least $-1-\sqrt{2}$, Linear Algebra Appl. {\bf 226--228}:577--591 (1995). \bibitem{T} T.~Taniguchi, On graphs with the smallest eigenvalue at least $-1-\sqrt{2}$, part I, Ars.~Math.~Comtemp. {\bf 1}:81--98 (2008). \bibitem{T2} T.~Taniguchi, On graphs with the smallest eigenvalue at least $-1-\sqrt{2}$, part II, in preparation. \end{thebibliography}

## Place

Location: Bled, Slovenia
Address: Best Western Hotel Kompas Bled

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