Circulants and Sequences

A graph G is stable if its normalized chromatic difference sequence is equal to the normalized chromatic di erence sequence of G G, the Cartesian product of G with itself. Let be the independence number of G and ! be its clique number. Suppose that G has n vertices. We show that the rst ! terms of the normalized chromatic di erence sequence of a stable graph G must be =n; and further that if G has odd girth 2k+ 1, then the rst three terms of its normalized chromatic di erence sequence are =n; =n; =n, where =k. We derive from this sequence an upper bound on the independence ratio of G, which agrees with the lower bound of Haggkvist for k = 2 and of Albertson, Chan and Haas for k 3. Zhou has shown that circulants and nite abelian Cayley graphs are stable. Let G be a circulant with symbol set S and n vertices. We say S = fa1; a2; : : : ; asg is reversible if a1 + as = a2 + as 1 = = ab s 2 c + ad s 2 c. We show that the independence ratio (G) (S), and that if S is reversible, then limn!1 (G) = (S). We conjecture that (G) = (S) for a reversible circulant with suÆciently many vertices.