Integral graph
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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.[2]
- The only cycle graphs that are integral are , , and .[2]
- If a graph is integral, then so is its complement graph; for instance, the complements of complete graphs, edgeless graphs, are integral. If two graphs are integral, then so is their Cartesian product and strong product;[2] for instance, the Cartesian products of two complete graphs, the rook's graphs, are integral.[3] Similarly, the hypercube graphs, as Cartesian products of any number of complete graphs , are integral.[2]
- The line graph of a regular integral graph is again integral. For instance, as the line graph of , the octahedral graph is integral, and as the complement of the line graph of , the Petersen graph is integral.[2]
- 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.[4]
