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 
16cell  
24cell  
Clebsch graph  
cubical graph  
cuboctahedral graph  
Desargues graph  
HallJanko graph  
Hoffman graph  
HoffmanSingleton 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: SpringerVerlag, pp.4551, 1974.
Sloane, N.J.A. Sequences 064731 A and A077027 in “The OnLine 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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