Asymptotic results regarding the number of walks in a graph Original Research Article

Let Wk denote the number of walks of length k(≥ 0) in a finite graph G, and define Δk = Δk(G) := Wk+1Wk−1 − Wk2. The condition Δ2k−1(G) > 0 is satisfied for all k ϵ View the MathML source, and for all graphs G except the harmonic graphs for which Δk = 0 holds for all k ≥ 2—and thus, in particular, for the regular graphs for which Δk = 0 holds for all k ϵ View the MathML source. In contrast, the sign of Δ2k(G) may be positive as well as negative, depending on k and G. We show that, for every finite graph G, there exist unique numbers N ϵ View the MathML source0, τ1, …, τN, a1, …, aN, b1, …, bN ϵ View the MathML source≥0 with 0 <- τ1 < τ2 < ⋯ < τN and a1 + b1, a2 + b2, …, aN + bN > 0 that can be computed in terms of the main eigenvalues and -angles of G such that View the MathML source holds for all k in View the MathML source. Consequently, the limits limk→∞View the MathML source and limk→∞View the MathML sourcealways exist and the sign of Δ2k is constant for all sufficiently large k.