Hoffman graphs and edge-signed graphs

Tetsuji Taniguchi
Hiroshima Institute of Technology



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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