Edge-Isoperimetric Problem for Cayley Graphs and Generalized Takagi Functions

Abstract. Let G be a finite abelian group of exponent m ≥ 2. For subsets A,S ⊆ G, denote by ∂S(A) the number of edges from A to its complement G \ A in the directed Cayley graph, induced by S on G. We show that if S generates G, and A is non-empty, then ∂S(A) ≥ e m |A| ln |G| |A| . Here the coefficient e = 2.718 . . . is best possible and cannot be replaced with a number larger than e. For homocyclic groups G of exponent m, we find an explicit closed-form expression for ∂S(A) in the case where S is the “standard” generating subset of G, and A is an initial segment of G with respect to the lexicographic order induced by S. Namely, we show that in this situation