On the Ádám Conjecture on Circulant Graphs

We investigate the condition for isomorphism between circulant graphs which is known as the Ádám property. We describe a wide class of graphs for which the Ádám conjecture holds (and even in a stronger form refering to isospectrality rather than to isomorphism of graphs. In the important case of double loop (circulant graphs of degree 4) we give an almost complete classification of all possible isospectral graphs. Such graphs arise in many applications including telecommunication networks, VLSI design and distributed computation and have been extensively studied in the literature. Our method is based on studying the graph spectra with the aid of some deep results of algebraic number theory. Dept of Computing, MPCE, Macquarie University, Sydney, NSW 2109, Australia. [email protected] Departimento di Matematica, Terza Università delgi Studi, Largo S. L. Murialdo, 1, Roma, 00146, Italy. [email protected] Dept of Computing, MPCE, Macquarie University, Sydney, NSW 2109, Australia. [email protected]