Geodesic
On \(k\)-walk-regular graphs
The electronic journal of combinatorics, Tome 16 (2009) no. 1
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Considering a connected graph $G$ with diameter $D$, we say that it is $k$-walk-regular, for a given integer $k$ $(0\leq k \leq D)$, if the number of walks of length $\ell$ between any pair of vertices only depends on the distance between them, provided that this distance does not exceed $k$. Thus, for $k=0$, this definition coincides with that of walk-regular graph, where the number of cycles of length $\ell$ rooted at a given vertex is a constant through all the graph. In the other extreme, for $k=D$, we get one of the possible definitions for a graph to be distance-regular. In this paper we show some algebraic characterizations of $k$-walk-regularity, which are based on the so-called local spectrum and predistance polynomials of $G$.
DOI : 10.37236/136
Classification : 05C12, 05C50, 05E30, 05C25
Mots-clés : algebraic characterizations of \(k\)-walk-regularity 
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     author = {C. Dalf\'o and M. A. Fiol and E. Garriga},
     title = {On \(k\)-walk-regular graphs},
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     year = {2009},
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     number = {1},
     doi = {10.37236/136},
     zbl = {1226.05107},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/136/}
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C. Dalfó; M. A. Fiol; E. Garriga. On \(k\)-walk-regular graphs. The electronic journal of combinatorics, Tome 16 (2009) no. 1. doi: 10.37236/136

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