Formulae derivation for numbers of fixed length cycles in rook's graphs

A technique for symbolic evaluation of the explicit formulae for counting fixed length cycles in undirected graphs is presented. Transformations of sums in the formulae are illustrated on the example of rook’s graphs associated with an NxN chessboard. By means of the explicit formulae for counting 3, 4,..., 7-cycles in arbitrary graphs, we derived the numbers of such cycles in rook’s graphs as polynomials in N.