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### Doubly regular asymmetric digraphs

Noboru Ito
1988 Discrete Mathematics
Namely D is regular of valency k and doubly regular of double valency A. 2 . 0 Proposition 5 . 205 Some basic properties of DRADs Proposition 1. v 2 2k + 1 and hence k 2 2A. + 1. Proof.  ...  Now we define a digraph D = (V, A), where V and A denote the sets of vertices and arcs respectively, as follows: V = P, and (a, b) is an arc if and only if b E T-'(a).  ...

### Normally Regular Digraphs [article]

Leif K Jørgensen
2014 arXiv   pre-print
A connected k-regular digraph with normal adjacency matrix is a normally regular digraph if and only if all eigenvalues other than k are on one circle in the complex plane.  ...  We prove that the adjacency matrix of a normally regular digraph is normal.  ...  Then the Cayley graph Cay(F, Q) is a doubly regular tournament. We conclude this section with two small asymmetric normally regular digraphs.  ...

### Hoffman's coclique bound for normal regular digraphs, and nonsymmetric association schemes [article]

2015 arXiv   pre-print
We extend Hoffman's coclique bound for regular digraphs with the property that its adjacency matrix is normal, and discuss cocliques attaining the inequality.  ...  As a consequence, we characterize skew-Bush-type Hadamard matrices in terms of digraphs. We present some normal digraphs whose vertex set is decomposed into disjoint cocliques attaining the bound.  ...  We will refer to a doubly regular asymmetric digraph with parameters (v, k, λ) as a DRAD(v, k, λ).  ...

### Page 652 of Mathematical Reviews Vol. , Issue 91B [page]

1991 Mathematical Reviews
Alan Tucker (1-SUNYS-S) 91b:05091 05C25 05B10 05C20 20B25 Ito, Noboru (J-PKNN-AM) Doubly regular asymmetric digraphs with rank 5 automorphism groups.  ...  In this paper the author classifies certain doubly regular asymmetric (v,k,A) diagraphs with rank 5 automorphism groups.  ...

### Page 8085 of Mathematical Reviews Vol. , Issue 2003k [page]

2003 Mathematical Reviews
T is asymmetric if whenever (x, y) is an arc then (y,x) is not an are. A doubly regular asymmetric digraph with parameters (v,k, A) is referred to as DRAD(v,k,A).  ...  .]; however this question is still open.” 2003k:05059 05C20 05B05 lonin, Yury J. (1-CMI; Mount Pleasant, MI); Kharaghani, Hadi (3-LETH; Lethbridge, AB) Doubly regular digraphs and symmetric designs.  ...

Bahman Gharesifard, Jorge Cortes
2010 Proceedings of the 2010 American Control Conference
In this paper, we fully characterize the class of digraphs that admit an edge weight assignment that makes the digraph adjacency matrix doubly stochastic.  ...  Digraphs with doubly stochastic adjacency matrices play an essential role in a variety of cooperative control problems including distributed averaging, optimization, and gossiping.  ...  G a C-regular digraph with C < ds(G).  ...

### Page 6172 of Mathematical Reviews Vol. , Issue 90K [page]

1990 Mathematical Reviews
A digraph DZ = (V,U) of order v is a doubly regular asymmetric (v,x,A) digraph if (i) DM is regular of valency k, (ii) D is doubly regular of double valency A, and (iii) D is asymmetric.  ...  Hemminger (1-VDB) 90k:05102 05C50 05C20 Ito, Noboru (J-PKNN-AM) On spectra of doubly regular asymmetric digraphs of RH-type. Graphs Combin. 5 (1989), no. 3, 229-234. Let v,x,A be positive integers.  ...

### Doubly regular digraphs and symmetric designs

2003 Journal of combinatorial theory. Series A
If an incidence matrix N of a symmetric design is such that N þ N t is a ð0; 1Þ matrix, then N is an adjacency matrix of a doubly regular asymmetric digraph, and vice versa.  ...  We construct several parametrically new infinite families of such digraphs. To carry on some of these constructions, we obtain an infinite family of skew balanced generalized weighing matrices. r  ...  to a doubly regular asymmetric digraph with parameters ðv; k; lÞ as a DRADðv; k; lÞ: For further information on these and more general digraphs see [9, 11]  ...

### Distributed Strategies for Generating Weight-Balanced and Doubly Stochastic Digraphs

Bahman Gharesifard, Jorge Cortés
2012 European Journal of Control
We refer to a digraph as doubly stochasticable (weight-balanceable) if it admits a doubly stochastic (weight-balanced) adjacency matrix.  ...  Weight-balanced and doubly stochastic digraphs are two classes of digraphs that play an essential role in a variety of coordination problems, including formation control, agreement, and distributed optimization  ...  In particular, we are indebted to a reviewer who suggested the formulation of the doubly stochastic weight assignment problem as a constrained maximum flow problem.  ...

### New constructions of Deza digraphs [article]

Dean Crnkovic, Hadi Kharaghani, Sho Suda, Andrea Svob
2020 arXiv   pre-print
Deza digraphs were introduced in 2003 by Zhang and Wang as directed graph version of Deza graphs, that also generalize the notion of directed strongly regular graphs.  ...  In this paper we give several new constructions of Deza digraphs. Further, we introduce twin and Siamese twin (directed) Deza graphs and construct several examples.  ...  Divisible design digraphs (see [2] ) are natural generalization of doubly regular asymmetric digraphs.  ...

### Distributed strategies for generating weight-balanced and doubly stochastic digraphs [article]

Bahman Gharesifard, Jorge Cortes
2011 arXiv   pre-print
We refer to a digraph as doubly stochasticable (weight-balanceable) if it admits a doubly stochastic (weight-balanced) adjacency matrix.  ...  Weight-balanced and doubly stochastic digraphs are two classes of digraphs that play an essential role in a variety of cooperative control problems, including formation control, distributed averaging,  ...  Then we refer to G = (V, E, A) as a C-regular digraph. Necessary and sufficient conditions for doubly stochasticability.  ...

### Divisible design Cayley digraphs [article]

Dean Crnkovic, Hadi Kharaghani, Andrea Svob
2019 arXiv   pre-print
Divisible design digraphs which can be obtained as Cayley digraphs are studied. A characterization of divisible design Cayley digraphs in terms of the generating sets is given.  ...  Further, we give several constructions of divisible design Cayley digraphs and classify divisible design Cayley digraphs on v < 27 vertices.  ...  Divisible design digraphs are natural generalization of doubly regular asymmetric di-graphs.  ...

### Page 847 of Mathematical Reviews Vol. 47, Issue 4 [page]

1974 Mathematical Reviews
A a which eorems, simplex d num- fore at », Ont.) 4838 s in the 1-Kalb- mono- 50, and ve that , Man.) 4839 ‘omatic regular c index . exists ) paper red the COMBINATORICS existence of a regular graph of  ...  The authors use doubly stochastic matrices to obtain some elementary properties of directed graphs.  ...

### Algorithmic Approach to Non-symmetric 3-class Association Schemes [chapter]

Leif K. Jørgensen
2009 Algorithmic Algebraic Combinatorics and Gröbner Bases
By Lemma 4, the directed graph is then a normally regular digraph with µ = λ. Such digraphs are also known as doubly regular asymmetric digraphs.  ...  Any doubly regular asymmetric digraph with automorphism group of rank 4 is a relation of a non-symmetric association scheme with 3 classes.  ...

### Page 5935 of Mathematical Reviews Vol. , Issue 89K [page]

1989 Mathematical Reviews
Toshimasa Watanabe (J-HROSE-A) 89k:05042 05C20 Ito, Noboru (J-PKNN-AM) Doubly regular asymmetric digraphs. Proceedings of the First Japan Conference on Graph Theory and Applications (Hakone, 1986).  ...  Obviously one can interpret such a design as an asymmetric digraph. These objects have been studied in the case of a Hadamard design, in which case the digraph is a tournament [see  ...
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