Weakly distance-regular digraphs

Comellas, Francesc; Fiol, M. A.; Gimbert, Joan; Riera, Margarida Mitjana

doi:10.1016/j.jctb.2003.07.003

Cited by 14 publications

(15 citation statements)

References 24 publications

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“…Then, G is a weakly distance‐regular digraph (according to Comellas et al. ) with distance polynomials

p_{0} = 1, p_{1} = x, p_{2} = x^{2} - 1, p_{3} = \frac{1}{d} x^{3} - x,

and Hoffman polynomial

0true H = \sum_{i = 0}^{3} p_{i} = \frac{1}{d} (x^{3} - x) + x^{2} + x

, as shown in .…”

Section: Mixed Bipartite Moore Graphsmentioning

confidence: 99%

“…■ Moreover, the minimum polynomial of is ( ) = 4 − 2 2 . Then, is a weakly distance-regular digraph (according to Comellas et al [4]) with distance polynomials 0 = 1, 1 = , 2 = 2 − 1, 3 = 1 3 − , and Hoffman polynomial = ∑ 3 =0 = 1 ( 3 − ) + 2 + , as shown in (19). For some other values of the diameter, we also have some families of mixed graphs that are asymptotically dense.…”

Section: The Case Of Diameter Threementioning

confidence: 99%

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On bipartite‐mixed graphs

Dalfó¹,

Fiol

López

2018

Journal of Graph Theory

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“…Then, G is a weakly distance‐regular digraph (according to Comellas et al. ) with distance polynomials

p_{0} = 1, p_{1} = x, p_{2} = x^{2} - 1, p_{3} = \frac{1}{d} x^{3} - x,

and Hoffman polynomial

0true H = \sum_{i = 0}^{3} p_{i} = \frac{1}{d} (x^{3} - x) + x^{2} + x

, as shown in .…”

Section: Mixed Bipartite Moore Graphsmentioning

confidence: 99%

Section: The Case Of Diameter Threementioning

confidence: 99%

On bipartite‐mixed graphs

Dalfó¹,

Fiol

López

2018

Journal of Graph Theory

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“…If we replace Γ + 1 (v) by Γ − 1 (v), in the definition of distance-regularity, we get a new family of digraphs called "weakly distance-regular digraphs". This concept was introduced by F. Comellas et al [9], as a generalization of distance-regular digraphs. In fact, distanceregular digraphs are precisely weakly distance-regular digraphs with normal adjacency matrices (a matrix A is normal if AA t = A t A, where A t is the transpose of A).…”

Section: Introductionmentioning

confidence: 99%

Minimum Cuts of Distance-Regular Digraphs

Ashkboos¹,

Omidi²,

Shafiei³

et al. 2017

Electron. J. Combin.

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“…In fact, distance-regular digraphs are normal weakly distance-regular digraphs. Also, in [6] it has been shown that a digraph Γ of diameter D is weakly distance-regular if, for each nonnegative integer ℓ ≤ D, the number a ℓ uv of walks of length ℓ from a vertex u to a vertex v only depends on their distance ∂(u, v).…”

Section: Introductionmentioning

confidence: 99%

A spectral excess theorem for normal digraphs

Omidi

2015

J Algebr Comb

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It is known that every distance-regular digraph is connected and normal. An interesting question is: when is a given connected normal digraph distance-regular? Motivated by this question first we give some characterizations of weakly distanceregular digraphs. Specially we show that whether a given connected digraph to be weakly distance-regular only depends on the equality for two invariants. Then we show that a connected normal digraph Γ with d + 1 distinct eigenvalues is distance-regular if and only if the simple excess (the ratio of the square of mean of the numbers of shortest paths between vertices at distance d to the mean of the numbers of vertices at distance d from every vertex, which is zero if d is greater than the diameter) is equal to the spectral excess (a number which can be computed from the spectrum of Γ). In fact, this result is a new variation (a simple variation) of the spectral excess theorem due to Fiol and Garigga for connected normal digraphs. Using these results we derive another variation (a weighted variation) of the spectral excess theorem for connected normal digraphs. Distance regularity of a digraph (also a graph) is in general not determined by its spectrum. For application of the simple variation we show that distance regularity of a connected normal digraph Γ (with d + 1 distinct eigenvalues) is determined by its spectrum and the invariant δ d (the mean of the numbers of vertices at distance d from every vertex). Finally as an application of the weighted variation we show that every connected normal digraph Γ with d + 1 distinct eigenvalues and diameter D is either a bipartite digraph, or a generalized odd graph or it has odd-girth at most min{2d − 1, 2D + 1}, generalizing a result of van Dam and Haemers and also a recent result of Lee and Weng.

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Weakly distance-regular digraphs

Cited by 14 publications

References 24 publications

On bipartite‐mixed graphs

On bipartite‐mixed graphs

Minimum Cuts of Distance-Regular Digraphs

A spectral excess theorem for normal digraphs

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