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## Characterizing (l, m)-walk-regular graphs

Author
Dalfo, C.; Fiol, M.; Garriga, E.
Type of activity
Journal article
Journal
Linear algebra and its applications
Date of publication
2010-12-30
Volume
433
Number
11-12
First page
1821
Last page
1826
DOI
https://doi.org/10.1016/j.laa.2010.06.042
Repository
http://hdl.handle.net/2117/3007
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.
Keywords
Adjacency matrix, Distance-regular graph, Predistance polynomial, Preintersection number, Spectrum, Walk-regular graph
Group of research
COMBGRAPH - Combinatorics, Graph Theory and Applications