# Integral graph

In the mathematical field of graph theory, an **integral graph** is a graph whose spectrum consists entirely of integers. In other words, a graphs is an integral graph if all the eigenvalues of its characteristic polynomial are integers.^{[1]}

The notion was introduced in 1974 by Harary and Schwenk.^{[2]}

## Examples

- The complete graph
*K*is integral for all_{n}*n*. - The edgeless graph is integral for all
*n*. - Among the cubic symmetric graphs the utility graph, the Petersen graph, the Nauru graph and the Desargues graph are integral.
- The Higman–Sims graph, the Hall–Janko graph, the Clebsch graph, the Hoffman–Singleton graph, the Shrikhande graph and the Hoffman graph are integral.

## References

- ↑ Weisstein, Eric W. "Integral Graph".
*MathWorld*. - ↑ Harary, F. and Schwenk, A. J. "Which Graphs have Integral Spectra?" In Graphs and Combinatorics (Ed. R. Bari and F. Harary). Berlin: Springer-Verlag, pp. 45–51, 1974.

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