TY - JOUR
T1 - Directed strongly regular graphs obtained from coherent algebras
AU - Klin, Mikhail
AU - Munemasa, Akihiro
AU - Muzychuk, Mikhail
AU - Zieschang, Paul Hermann
N1 - Funding Information:
This paper is a revised and shortened version of the preprint [29]. This preprint stimulated a new wave of interest in the investigations of d.s.r.g.’s, in particular results by L. Jorgensen, S. A. Hobart and T.J. Shaw, A. Duval and D. Iourinski. We refer to [11,21,25,30] where a part of recent investigations is discussed with more details. We also mention that the most of the current status of the theory of d.s.r.g.’s can be found in Andries Brouwer’s home page http://www.cwi.nl/∼aeb/math/dsrg/. ∗Corresponding author. E-mail address: [email protected] (M. Klin). 1Partially supported by DAAD allowance for study visit to Germany, by the Israeli Ministry of Absorption and by the Department of Mathematical Sciences, University of Delaware. 2 Supported by the research grant no. 3869 from the Israeli Ministry of Science.
PY - 2004/1/15
Y1 - 2004/1/15
N2 - The notion of a directed strongly regular graph was introduced by A. Duval in 1988 as one of the possible generalizations of classical strongly regular graphs to the directed case. We investigate this generalization with the aid of coherent algebras in the sense of D.G. Higman. We show that the coherent algebra of a mixed directed strongly regular graph is a non-commutative algebra of rank at least 6. With this in mind, we examine the group algebras of dihedral groups, the flag algebras of a Steiner 2-designs, in search of directed strongly regular graphs. As a result, a few new infinite series of directed strongly regular graphs are constructed. In particular, this provides a positive answer to a question of Duval on the existence of a graph with certain parameter set having 20 vertices. One more open case with 14 vertices listed in Duval's paper is ruled out, while new interpretations in terms of coherent algebras are given for many of Duval's results.
AB - The notion of a directed strongly regular graph was introduced by A. Duval in 1988 as one of the possible generalizations of classical strongly regular graphs to the directed case. We investigate this generalization with the aid of coherent algebras in the sense of D.G. Higman. We show that the coherent algebra of a mixed directed strongly regular graph is a non-commutative algebra of rank at least 6. With this in mind, we examine the group algebras of dihedral groups, the flag algebras of a Steiner 2-designs, in search of directed strongly regular graphs. As a result, a few new infinite series of directed strongly regular graphs are constructed. In particular, this provides a positive answer to a question of Duval on the existence of a graph with certain parameter set having 20 vertices. One more open case with 14 vertices listed in Duval's paper is ruled out, while new interpretations in terms of coherent algebras are given for many of Duval's results.
KW - Automorphism group
KW - Building
KW - Coherent algebra
KW - Dihedral group
KW - Directed strongly regular graph
KW - Doubly regular tournament
KW - Flag algebra
KW - Permutation group
KW - Steiner system
UR - http://www.scopus.com/inward/record.url?scp=0242271793&partnerID=8YFLogxK
U2 - 10.1016/j.laa.2003.06.020
DO - 10.1016/j.laa.2003.06.020
M3 - Article
AN - SCOPUS:0242271793
SN - 0024-3795
VL - 377
SP - 83
EP - 109
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
IS - 1-3
ER -