Integral graph
In the mathematical field of graph theory, an integral graph is a graph whose adjacency matrix's spectrum consists entirely of integers. In other words, a graph is an integral graph if all of the roots of the characteristic polynomial of its adjacency matrix are integers.[1]
The notion was introduced in 1974 by Frank Harary and Allen 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.
- The Sudoku graphs, graphs whose vertices represent cells of a Sudoku board and whose edges represent cells that should not be equal, are integral.[3]
- Rook's graphs are integral.[4]
References
- ^ Weisstein, Eric W., "Integral Graph", MathWorld
- ^ Harary, Frank; Schwenk, Allen J. (1974), "Which graphs have integral spectra?", in Bari, Ruth A.; Harary, Frank (eds.), Graphs and Combinatorics: Proceedings of the Capital Conference on Graph Theory and Combinatorics at the George Washington University, Washington, D.C., June 18–22, 1973, Lecture Notes in Mathematics, vol. 406, Springer, pp. 45–51, doi:10.1007/BFb0066434, MR 0387124
- ^ Sander, Torsten (2009), "Sudoku graphs are integral", Electronic Journal of Combinatorics, 16 (1): Note 25, 7, MR 2529816
- ^ Doob, Michael (1970), "On characterizing certain graphs with four eigenvalues by their spectra", Linear Algebra and its Applications, 3: 461–482, doi:10.1016/0024-3795(70)90037-6, MR 0285432