A Solution of the Isomorphism Problem for Circulant Graphs

All graphs considered in the paper are directed. Let % be a graph on n vertices which we identify with the elements of the additive cyclic group Zn 1⁄4 f0; 1; . . . ; n 1g. The graph % is called circulant if it has a cyclic symmetry, that is, if the permutation ð0; 1; 2; . . . ; n 1Þ is an automorphism of the graph. Each circulant graph is completely determined by its connection set S which is the set of all points connected to 0 in %. Two vertices i and j are %-connected if and only if i j 2 S where the di1erence is taken in Zn. Thus a circulant graph is a Cayley graph over the cyclic group Zn. In what follows we identify a circulant graph with its set of arcs which will be denoted as CayðZn; SÞ, that is, CayðZn; SÞ :1⁄4 fðx; yÞ jx y 2 Sg where S Zn is the connection set of the graph. If f 2 SymðZnÞ, then CayðZn; SÞ is de4ned as fðx ; yÞ j ðx; yÞ 2 CayðZn; SÞg. Two circulant graphs CayðZn; SÞ and CayðZn; T Þ are isomorphic if there exists f 2 SymðZnÞ such that CayðZn; SÞ 1⁄4 CayðZn; T Þ. The isomorphism problem for circulant graphs is stated as follows.