A note on the invariant factors of the walk matrix of a graph

Abstract For a given graph G with n vertices, the walk matrix W of G is defined as W = [ e A e A 2 e ⋯ A n − 1 e ] , where A is the adjacency matrix of the graph G and e is the vector of all ones. In this paper, we prove that for any positive integer k, at most ⌊ n 2 ⌋ invariant factors of W are congruent to 2 k modulo 2 k + 1 .