On $(l,m)$-Walk-Regular Digraphs

Abstract

In this paper, we introduce a class of digraphs called $(l,m)$-walk-regular digraphs, a common generalization of both weakly distance-regular digraphs $[1]$ and $k$-walk-regular digraphs $[3]$, and give several characterizations of them about their regularity properties that are related to distance and about the number of walks of given length between vertices at a given distance.