Definite Integral Graph – Integrals — Learn Desmos

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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

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, 2, … nodes are 0, 2, 3, 6, 10, 20, 33, 71, … (OEIS A077027), illustrated above for small

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Đang xem: Integral graph

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The numbers of connected simple integral graphs on

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, 2, … nodes are 1, 1, 1, 2, 3, 6, 7, 22, 24, 83, … (OEIS A064731), illustrated above for small

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

graph integral for

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of the form

antiprism graph 3
complete graph

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all
cycle graph

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2, 3, 4, 6
empty graph all
prism graph 3, 4, 6
star graph

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*
wheel graph

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4

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

graph graph spectrum
16-cell
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24-cell
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Clebsch graph
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cubical graph
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cuboctahedral graph
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Desargues graph
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Hall-Janko graph
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Hoffman graph
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Hoffman-Singleton graph
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Levi graph
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M22 graph
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McLaughlin graph
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octahedral graph
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pentatope
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Petersen graph
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Shrikhande graph
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*

-simplex

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small triakis octahedral graph
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Sylvester graph
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tesseract
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tetrahedral graph
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truncated tetrahedral graph
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utility 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.”

CITE THIS AS:

Weisstein, Eric W. “Integral Graph.” Fromwnyrails.org–A wnyrails.org Web Resource. https://wnyrails.org/IntegralGraph.html

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