Integral graph
In the mathematical field of graph theory, an integral graph is a graph whose spectrum consists entirely of integers. In other words, a graph is an integral graph if all of the roots 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.
- A regular graph is periodic if and only if it is an integral graph.
- A walk-regular graph that admits perfect state transfer is an integral graph.
References
- ^ Weisstein, Eric W. "Integral Graph". MathWorld.
- ^ 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.