Directed strongly regular graphs with μ=λ

Jørgensen, Leif K.

doi:10.1016/s0012-365x(00)00325-3

Cited by 22 publications

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“…The groups are non-abelian. In fact, in [9] we observed that it follows from a theorem by Klin, Munemasa, Muzychuk and Zieschang [11] that a directed strongly regular graph with 0 < t < k cannot be a Cayley graph of an abelian group. A nice proof of this result was found by Lyubshin and Savchenko [13].…”

Section: Proposition 2 the Kautz Digraph L(k Q ) Is A Cayley Graph Imentioning

confidence: 95%

New mixed Moore graphs and directed strongly regular graphs

Jørgensen

2015

Discrete Mathematics

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a b s t r a c tA directed strongly regular graph with parameters (n, k, t, λ, µ) is a k-regular directed graph with n vertices satisfying that the number of walks of length 2 from a vertex x to a vertex y is t if x = y, λ if there is an edge directed from x to y and µ otherwise. If λ = 0 and µ = 1 then we say that it is a mixed Moore graph. It is known that there are unique mixed Moore graphs with parameters (k 2 + k, k, 1, 0, 1), k ≥ 2, and (18, 4, 3, 0, 1). We construct a new mixed Moore graph with parameters (108, 10, 3, 0, 1) and also new directed strongly regular graphs with parameters (36, 10, 5, 2, 3) and (96, 13, 5, 0, 2). This new graph on 108 vertices can also be seen as an example of a so called multipartite Moore digraph. Finally we consider the possibility that mixed Moore graphs with other parameters could exist, in particular the first open case which is (40, 6, 3, 0, 1).

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Section: Proposition 2 the Kautz Digraph L(k Q ) Is A Cayley Graph Imentioning

confidence: 95%

New mixed Moore graphs and directed strongly regular graphs

Jørgensen

2015

Discrete Mathematics

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“…Hence the parameters are as in the statement of Lemma 4.10, which is a contradiction. If = 4 we obtain U = 1, and the parameters of the PSQ are n = 16, r = 2, q = 4, k = 6, μ = 1, λ = 1, t = 5, but no PSQ corresponding to this set of parameters exist; in fact, no DSRG with these parameters exists (see [12]). Therefore, Situation 2 is impossible, and we are left with Situation 1.…”

Section: (417)mentioning

confidence: 99%

“…In 1997 Klin, Munemasa, Muzychuk and Zieschang [15] proved that a DSRG cannot be a Cayley digraph of an abelian group, whereas in 1999, Hobart and Shaw [11] gave constructions of DSRGs which are Cayley digraphs of nonabelian groups. In 2002 Fiedler, Klin and Muzychuk [8] determined DSRGs of order v 20 having a vertex-transitive automorphism group, which combined together with results in [12] gives a complete answer to Duval's question posed in [5] about the existence of DSRGs of order v 20. Further examples of Cayley digraphs which are DSRGs were given in 2003 by Duval and Iourinski [6].…”

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confidence: 99%

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Partial sum quadruples and bi-Abelian digraphs

Araluze

Kovács

Kutnar

et al. 2012

Journal of Combinatorial Theory, Series A

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As a generalization of undirected strongly regular graphs, a digraph X without loops, of valency k and order v is said to be a (v, k, μ, λ,t)-directed strongly regular graph whenever for any vertex u of X there are t undirected edges having u as an endvertex and for every two different vertices u and w of X the number of paths of length 2 starting at u and ending at w is λ or μ depending only on whether uw is an arc of X or not. An m-Cayley digraph of a group H is a digraph admitting a semiregular group of automorphisms having m orbits, all of equal length, isomorphic to H. In this paper, the structure of directed strongly regular 2-Cayley graphs of cyclic groups is investigated. In particular, the arithmetic conditions on parameters v, k, μ, λ, and t are given.Also, several infinite families of directed strongly regular graphs which are also 2-Cayley digraphs of abelian groups are constructed.

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“…Interest in these graphs was recently revived by Klin, Munemasa, Muzychuk, and Zieschang [9], and there have been a number of recent papers ( [3], [4], [6], [7], [8]). …”

Section: Introductionmentioning

confidence: 99%

Strongly regular graphs

Godsil¹,

Meagher²

2015

Erdős–Ko–Rado Theorems: Algebraic Approaches

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Acknowledgment: The second and third authors would like to thank the department of Combinatorics and Optimization at the University of Waterloo for their hospitality during the initial research for this paper. AbstractWe develop a theory of representations in R m for directed strongly regular graphs, which gives a new proof of a nonexistence condition of Jørgensen [8]. We also describe some new constructions.

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Directed strongly regular graphs with μ=λ

Cited by 22 publications

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New mixed Moore graphs and directed strongly regular graphs

New mixed Moore graphs and directed strongly regular graphs

Partial sum quadruples and bi-Abelian digraphs

Strongly regular graphs

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