Distance-transitive and distance-regular digraphs

Damerell, R. M.

doi:10.1016/s0095-8956(81)80009-3

Cited by 34 publications

(26 citation statements)

References 3 publications

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“…Let us consider the vertices u ¼ x 1 x 2 yx D and v ¼ x 2 yx D x 1 : Clearly distðu; vÞ ¼ 1 but distðv; uÞ41 if D41: Hence, the digraph is not stable. (The above can also be proved directly from the definition of strongly distance-regular digraph given in [9].) Given an integer xX2; let us denote by P x the set of (ordered) t-tuples ðx 1 ; x 2 ; y; x t Þ; which are a partition of x; x 1 þ x 2 þ ?…”

Section: The Cycle Prefix Digraphsmentioning

confidence: 98%

“…where a k k a0: Conversely, let us assume that there are constants a k l ; with a k k a0; satisfying (9). This implies that Eqs.…”

Section: Weak Distance-regularitymentioning

confidence: 99%

“…(In fact, notice that (17) just counts in two ways the number of ordered triples ðx; y; zÞ such that distðx; yÞ ¼ k; distðx; zÞ ¼ i; and distðz; yÞ ¼ j:) & Thus, any distance-regular digraph G satisfies all the above properties of weakly distance-regular digraphs. Moreover, since G is stable and has girth gAf2; D; D þ 1g; see [9], we have…”

Section: Article In Pressmentioning

confidence: 99%

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Weakly distance-regular digraphs

Comellas

Fiol

Gimbert

et al. 2004

Journal of Combinatorial Theory, Series B

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We introduce the concept of weakly distance-regular digraph and study some of its basic properties. In particular, the (standard) distance-regular digraphs, introduced by Damerell, turn out to be those weakly distance-regular digraphs which have a normal adjacency matrix. As happens in the case of distance-regular graphs, the study is greatly facilitated by a family of orthogonal polynomials called the distance polynomials. For instance, these polynomials are used to derive the spectrum of a weakly distance-regular digraph. Some examples of these digraphs, such as the butterfly and the cycle prefix digraph which are interesting for their applications, are analyzed in the light of the developed theory. Also, some new constructions involving the line digraph and other techniques are presented. r

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Section: The Cycle Prefix Digraphsmentioning

confidence: 98%

“…where a k k a0: Conversely, let us assume that there are constants a k l ; with a k k a0; satisfying (9). This implies that Eqs.…”

Section: Weak Distance-regularitymentioning

confidence: 99%

Section: Article In Pressmentioning

confidence: 99%

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Weakly distance-regular digraphs

Comellas

Fiol

Gimbert

et al. 2004

Journal of Combinatorial Theory, Series B

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“…A method to calculate E(G,Q(i)) is provided. Section 3 deals with the distanceregular digraphs of girth 4 (see [3,5] for definitions and details). The only known examples of such digraphs were constructed in [5,6] groups.…”

Section: E(g K) Of Y'g X When 8qg X Is Determined and E(g K)mentioning

confidence: 99%

“…See [3,5] for the definitions and the properties of distance-regular digraphs. Our concern here is the distance-regular digraphs of girth 4, which are the digraphs with adjacency matrix A such that {I,A,B = J-I-A-At, A '} is a P-polynomial commutative association scheme.…”

Section: The Abelian Singer Groups Of the Distance-regular Digraphs Omentioning

confidence: 99%

On theG-matrices with entries and eigenvalues inQ(i)

Hou

1992

Graphs and Combinatorics

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Abstract. Let G be a finite abelian group, K a subfield of C, C[G] regarded as an algebra of matrices. ~1~ = {A e C [C]L all the entries and eigenvalues of A are in K} is an association algebra over K. In this paper, the association scheme of .d~ is determined and in the case K = Q(i), the first eigenmatrix of the association scheme computed. As an application, it is proved that Z4 x Z4 • Z4 is the only abelian group admitted as a Singer group by some distance-regular digraph of girth 4 on 64 vertices.

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Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics

Bannaĭ

1990

Orthogonal Polynomials

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ABSTRACT. This paper surveys the role of orthogonal polynomials in Algebraic Combinatorics, an area which includes association schemes, coding theory, design theory, various theories of group representation, and so on. The main topics discussed in this paper include the following: The connection between orthogonal polynomials and P -polynomial (or Q -polynomial) association schemes. The classification problem for P -and Q -polynomial association schemes and its connection with Askey-Wilson orthogonal polynomials. Delsarte theory of codes and designs in association schemes. The nonexistence of perfect e -codes and tight t -designs through the study of the zeros of orthogonal polynomials. The possible importance of multi-variable versions of Askey-Wilson polynomials in the future study of general commutative association schemes.

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Distance-transitive and distance-regular digraphs

Cited by 34 publications

References 3 publications

Weakly distance-regular digraphs

Weakly distance-regular digraphs

On theG-matrices with entries and eigenvalues inQ(i)

Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics

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