A Property of Walks in Graphs

The aim of this note is to call attention to a peculiar odd–even regularity regarding the number of walks in a finite graph G: Let Wk denote the number of walks of length k (≥ 0) in G and put ∆k := Wk+1Wk−1−W 2 k (k ≥ 1) . Then • ∆2k−1 ≥ 0 holds for all k ≥ 1, • one has either ∆1 ≥ 0 and ∆k = 0 for all k ≥ 2 which holds if and only if G is harmonic (i.e. if and only if the degree map which associates to every vertex v its degree degG(v) is an “eigenvector” of G) or one has ∆2k−1 > 0 for all k ≥ 1 in which case one has either ∆2k = 0 for all k ≥ 1 or there exists a constant k0 with ∆2k∆2k+2 > 0 for all k > k0, • there exist graphs with ∆2k > 0 and graphs with ∆2k < 0 for all k ≥ 1, and • no graphs have been found yet with ∆2k−1 > 0 and ∆2k = 0 for all k ≥ 1, or with ∆2k∆2k′ < 0 for two distinct integers k and k′ ≥ 1.