On the Smith normal form of walk matrices

Let G be a graph with n vertices. The walk matrix W(G) of G is the matrix [e,Ae,…,An−1e], where A is the adjacency matrix of G and e is the all-one vector. Let W be a walk matrix of order n. We show that at most [Formula presented] invariant factors of W are congruent to 2 modulo 4. As a consequence, it is proved that, for any n×n walk matrix W with 2-rank r, the determinant of W is always a multiple of [Formula presented]. Moreover, if [Formula presented] is odd and square-free, then the Smith normal form of W can be recovered uniquely from the triple (n,r,det⁡W). © 2020 Elsevier Inc.