# Hoffman graphs and edge-signed graphs

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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}