Truncation error (numerical integration): Difference between revisions
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[ | In the [[mathematics|mathematical]] field of [[graph theory]], an '''integral graph''' is a graph whose [[Spectral graph theory|spectrum]] consists entirely of integers. In other words, a graphs is an integral graph if all the [[eigenvalues]] of its [[characteristic polynomial]] are integers.<ref>{{MathWorld|urlname=IntegralGraph|title=Integral Graph}}</ref> | ||
The notion was introduced in 1974 by Harary and Schwenk.<ref>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.</ref> | |||
==Examples== | |||
*The [[complete graph]] ''K<sub>n</sub>'' is integral for all ''n''. | |||
*The [[edgeless graph]] <math>\bar K_n</math> 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== | |||
{{reflist}} | |||
[[Category:Graph families]] | |||
[[Category:Algebraic graph theory]] | |||
Revision as of 04:00, 31 January 2014
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 Kn is integral for all 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
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