Целый граф

Целый граф (целочисленный граф) — граф, спектр матрицы смежности (инвариант графа) которого состоит полностью из целых чисел. Другими словами, граф является целым графом, при условии, что все корни характеристического многочлена его матрицы смежности являются целыми числами^[1]. Понятие ввели в 1974 году Харари и Швенк^[2].

Примеры:

полный граф $K_{n}$ является целым для всех $n$ ;
граф без рёбер ${\bar {K}}_{n}$ является целым для всех $n$ ;
среди кубических симметричных графов целыми являются коммунальный граф, граф Петерсена, граф Науру и граф Дезарга;
целыми являются также граф Хигмана — Симса, граф Холла — Янко, граф Клебша, граф Хоффмана — Синглтона, граф Шрикханде и граф Хоффмана, Граф Госсета;
графы судоку, вершины которых представляют ячейки поля Судоку, а рёбра представляют ячейки, которые не должны быть равны, являются целыми графами^[3].

Регулярный граф является периодическим^[англ.] тогда и только тогда, когда он целый. Граф регулярных блужданий, удовлетворяющий условиям идеальной передачи квантового состояния^[англ.], является целым графом.

Примечания

↑ Weisstein, Eric W. Integral Graph (англ.) на сайте Wolfram MathWorld.
↑ Harary F., Schwenk A. J. Which Graphs have Integral Spectra? // Graphs and Combinatorics / R. Bari и F. Harary. — Berlin: Springer-Verlag, 1974. — С. 45—51.
↑ Torsten Sander. Sudoku graphs are integral // Electronic Journal of Combinatorics. — 2009. — Т. 16, вып. 1. — С. Note 25, 7. Архивировано 15 апреля 2011 года.

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