# Definite Integral Graph – Integrals — Learn Desmos

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

.

Đang xem: Integral graph

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.

 graph graph spectrum 16-cell 24-cell Clebsch graph cubical graph cuboctahedral graph Desargues graph Hall-Janko graph Hoffman graph Hoffman-Singleton graph Levi graph M22 graph McLaughlin graph octahedral graph pentatope Petersen graph Shrikhande graph -simplex small triakis octahedral graph Sylvester graph tesseract tetrahedral graph truncated tetrahedral graph utility graph

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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