# Hoffman graphs and edge-signed graphs

### Tetsuji Taniguchi Hiroshima Institute of Technology

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Minisymposium: SPECTRAL GRAPH THEORY

Content: Hoffman graphs were introduced by Woo and Neumaier [2] to extend the results of Hoffman [1]. Hoffman proved what we would call Hoffman's limit theorem which asserts that, in the language of Hoffman graphs, the smallest eigenvalue of a fat Hoffman graph is a limit of the smallest eigenvalues of a sequence of ordinary graphs with increasing minimum degree. Woo and Neumaier [2] gave a complete list of fat indecomposable Hoffman graphs with smallest eigenvalue at least $-1-\sqrt{2}$. Moreover they [2, Open Problem 4] raised the problem of classifying fat Hoffman graphs with smallest eigenvalue at least $-3$. Therefore we study the problem. In the process, we noticed that it is necessary to know the details of special graphs of Hoffman graphs. There are deep relations between Hoffman graph" and Edge-signed graph". In this talk, we introduce Hoffman graphs and edge-signed graphs. Moreover we talk about some results and problems. \vspace{5mm} \noindent{\bf References:} \begin{itemize} \item[{[1]}] A.~J.~Hoffman, On graphs whose least eigenvalue exceeds $-1-\sqrt{2}$, Linear Algebra Appl. 16 (1977), 153--165. \item[{[2]}] R.~Woo and A.~Neumaier, On graphs whose smallest eigenvalue is at least $-1-\sqrt{2}$, Linear Algebra Appl. 226-228 (1995), 577--591 . \end{itemize}

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