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# Integral Graph

An integral graph, not to be confused with an integral embedding of a graph, is defined as a graph whose graph spectrum consists entirely of integers. The notion was first introduced by Harary and Schwenk (1974). The numbers of simple integral graphs on , 2, ... nodes are 0, 2, 3, 6, 10, 20, 33, 71, ... (OEIS A077027), illustrated above for small .

The numbers of connected simple integral graphs on , 2, ... nodes are 1, 1, 1, 2, 3, 6, 7, 22, 24, 83, ... (OEIS A064731), illustrated above for small .

The following table lists common graph classes and the their members which are integral.

 graph integral for of the form antiprism graph 3 complete graph all cycle graph 2, 3, 4, 6 empty graph all prism graph 3, 4, 6 star graph wheel graph 4

The following table lists some special named graphs that are integral and gives their spectra.

Eigenvalue, Graph Spectrum, Unit-Distance Graph

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## References

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.Sloane, N. J. A. Sequences 064731 A and A077027 in "The On-Line Encyclopedia of Integer Sequences."

Integral Graph

## Cite this as:

Weisstein, Eric W. "Integral Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IntegralGraph.html