On $k$-Walk-Regular Graphs
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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 $G$.
1986 ◽
Vol 41
(2)
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pp. 193-210
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2019 ◽
Vol 12
(07)
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pp. 2050009
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2018 ◽
Vol 34
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pp. 459-471
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2020 ◽
Vol 2020
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pp. 1-6
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2000 ◽
Vol 9
(6)
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pp. 573-585
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