On k-Walk-Regular Graphs
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Cita com:
hdl:2117/2237
Tipus de documentArticle
Data publicació2008-08
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Abstract
Considering a connected graph $G$ with diameter $D$, we say that it
is \emph{$k$-walk-regular}, for a given integer $k$ $(0\leq k \leq
D)$, if the number of walks of length $\ell$ between vertices $u$
and $v$ 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 present some algebraic characterizations of
$k$-walk-regularity, which are based on the so-called local spectrum
and predistance polynomials of $G$. Moreover, some results, concerning some parameters of a geometric nature, such as the cosines, and the spectrum of walk-regular graphs are presented.
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