Characterizing (l,m)-walk-regularity

Abstract

A graph $\G$ with diameter $D$ and $d+1$ distinct eigenvalues is said to be {\it $(\ell,m)$-walk-regular}, for some integers $\ell\in[0,d]$ and $m\in[0,D]$, $\ell\ge m$, if the number of walks of length $i\in [0,\ell]$ between any pair of vertices at distance $j\in [0,m]$ depends only on the values of $i$ and $j$. In this paper we study some algebraic and combinatorial characterizations of $(\ell,m)$-walk-regularity based on the so-called predistance polynomials and the preintersection numbers.