Closed walks and eigenvalues of Abelian Cayley graphs

We show that Abelian Cayley graphs contain many closed walks of even length. This implies that given k⩾3, for each ϵ>0, there exists C=C(ϵ,k)>0 such that for each Abelian group G and each symmetric subset S of G with 1∉S, the number of eigenvalues λi of the Cayley graph X=X(G,S) such that λi⩾k−ϵ is at least C⋅|G|. This can be regarded as an analogue for Abelian Cayley graphs of a theorem of Serre for regular graphs. To cite this article: S.M. Cioabă, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

RésuméSoit k⩾3, pour chaque ϵ>0, il existe une constante positive C=C(ϵ,k)>0 telle que pour chaque groupe abélien G et pour chaque sous-ensemble symétrique S⊂G ne contenant pas 1, le nombre de valeurs propres λi de graphe de Cayley X=X(G,S) qui satisfont λi⩾k−ϵ est au moins C⋅|G|. Pour citer cet article : S.M. Cioabă, C. R. Acad. Sci. Paris, Ser. I 342 (2006).