229 research outputs found
Notes on Various Methods for Constructing Directed Strongly Regular Graphs
Duval, in "A Directed Graph Version of Strongly Regular Graphs" [{\it Journal
of Combinatorial Theory}, Series A 47 (1988) 71 - 100], introduced the concept
of directed strongly regular graphs. In this paper we construct several rich
families of directed strongly regular graphs with new parameters. Our
constructions yielding new parameters are based on extending known explicit
constructions to cover more parameter sets. We also explore some of the links
between Cayley graphs, block matrices and directed strongly regular graphs with
certain parameters. Directed strongly regular graphs which are also Cayley
graphs are interesting due to their having more algebraic structure. We
construct directed strongly regular Cayley graphs with parameters
where and are integers with and
. We also give a new block matrix characterization for directed
strongly regular graphs with parameters , which were
first dicussed by Duval et al. in "Semidirect Product Constructions of Directed
Strongly Regular Graphs" [{\it Journal of Combinatorial Theory}, Series A 104
(2003) 157 - 167]
The application of representation theory in directed strongly regular graphs
The concept of directed strongly regular graphs (DSRG) was introduced by
Duval in 1988 \cite{A}.In the present paper,we use representation theory of
finite groups in order to investigate the directed strongly regular Cayley
graphs.We first show that a Cayley graph is not a directed
strongly regular graph if is a union of some conjugate classes of .This
generalizes an earlier result of Leif K.J{\o}rgensen \cite{J1} on abelian
groups.Secondly,by using induced representations,we have a look at the Cayley
graph with and
,determining its characteristic polynomial and its minimal
polynomial.Based on this result,we generalize the semidirect product method of
Art M. Duval and Dmitri Iourinski in \cite{D} and obtain a larger family of
directed strongly regular graphs.Finally,we construct some directed strongly
regular Cayley graphs on dihedral groups,which partially generalize the earlier
results of Mikhail Klin,Akihiro Munemasa,Mikhail Muzychuk,and Paul Hermann
Zieschang in \cite{K1}.By using character theory,we also give the
characterization of directed strongly regular Cayley graphs
with .Comment: 27 page
The Constructions of directed strongly regular graph by algebraic method
The concept of directed strongly regular graphs (DSRG) was introduced by
Duval in "A Directed Graph Version of Strongly Regular Graphs" [Journal of
Combinatorial Theory, Series A 47(1988)71-100]. Duval also provided several
construction methods for directed strongly regular graphs. In this paper, We
construct several new classes of directed strongly regular graphs which are
obtained by using Kronecker matrix product, Semidirect product and Cayley coset
graph. At the same time, using group representation, for two special cases, we
give some other sufficient and necessary conditions of Cayley graphs to be
DSRG. At last, we finish this paper with a discussion of some propositions of
in(out)-neighbours and automorphism group in directed strongly regular graphs.Comment: 35 pages,2 figure
Two Kinds of Constructions of Directed Strongly Regular Graphs from Partial Sum Families and Semi-direct Products of Groups
In this paper, we construct directed strongly regular graphs with new
parameters by using partial sum families with local rings. 16 families of new
directed strongly regular graphs are obtained and the uniform partial sum
families are given. Based on the cyclotomic numbers of finite fields, we
present two infinite families of directed strongly regular Cayley graphs from
semi-direct products of groups.Comment: 16 pages, 0 figure
Construction of directed strongly regular graphs using finite incidence structures
We use finite incident structures to construct new infinite families of
directed strongly regular graphs with parameters for integers and
(), and
for all prime powers and . The new graphs given by
these constructions have parameters , ,
, , and listed as feasible parameters on "Parameters of directed
strongly regular graphs," at by S. Hobart and A. E. Brouwer. We review these
constructions and show how our methods may be used to construct other infinite
families of directed strongly regular graphs.Comment: 13 page
Existence of unimodular triangulations - positive results
Unimodular triangulations of lattice polytopes arise in algebraic geometry,
commutative algebra, integer programming and, of course, combinatorics.
In this article, we review several classes of polytopes that do have
unimodular triangulations and constructions that preserve their existence.
We include, in particular, the first effective proof of the classical result
by Knudsen-Mumford-Waterman stating that every lattice polytope has a dilation
that admits a unimodular triangulation. Our proof yields an explicit (although
doubly exponential) bound for the dilation factor.Comment: 89 pages; changes from v2 and v1: the survey part has been expanded,
in particular the section on open question
Skew products and crossed products by coactions
Given a labeling c of the edges of a directed graph E by elements of a
discrete group G, one can form a skew-product graph E cross_c G. We show, using
the universal properties of the various constructions involved, that there is a
coaction delta of G on C*(E) such that C*(E cross_c G) is isomorphic to the
crossed product C*(E) cross_delta G. This isomorphism is equivariant for the
dual action deltahat and a natural action gamma of G on C*(E cross_c G);
following results of Kumjian and Pask, we show that C*(E cross_c G) cross_gamma
G is isomorphic to C*(E cross_c G) cross_{gamma,r} G, which in turn is
isomorphic to C*(E) tensor K(l^2(G)), and it turns out that the action gamma is
always amenable. We also obtain corresponding results for r-discrete groupoids
Q and continuous homomorphisms c: Q -> G, provided Q is amenable. Some of these
hold under a more general technical condition which obtains whenever Q is
amenable or second-countable.Comment: 22 pages, LaTeX2e, uses pb-diagram.st
Directed Strongly Regular Cayley Graphs on Dihedral groups
In this paper,we construct some directed strongly regular Cayley graphs on
dihedral groups,these generalizes some earlier constructions.We also
characterize some certain directed strongly regular Cayley graphs on dihedral
groups ,where is a prime and is a
positive integer.Comment: 20 page
Topological finiteness properties of monoids. Part 2: special monoids, one-relator monoids, amalgamated free products, and HNN extensions
We show how topological methods developed in a previous article can be
applied to prove new results about topological and homological finiteness
properties of monoids. A monoid presentation is called special if the
right-hand side of each relation is equal to . We prove results which relate
the finiteness properties of a monoid defined by a special presentation with
those of its group of units. Specifically we show that the monoid inherits the
finiteness properties and from its group of units. We also obtain
results which relate the geometric and cohomological dimensions of such a
monoid to those of its group of units. We apply these results to prove a
Lyndon's Identity Theorem for one-relator monoids of the form . In particular we show that all such monoids are of type
(and ), and that when is not a proper power, then
the monoid has geometric and cohomological dimension at most . The first of
these results resolves an important case of a question of Kobayashi from 2000
on homological finiteness properties of one-relator monoids. We also show how
our topological approach can be used to prove results about the closure
properties of various homological and topological finiteness properties for
amalgamated free products and HNN-extensions of monoids. To prove these results
we introduce new methods for constructing equivariant classifying spaces for
monoids, as well as developing a Bass-Serre theory for free constructions of
monoids.Comment: 36 pages, Major revision: final section extracted as a separate short
not
New infinite families of directed strongly regular graphs via equitable partitions
In this paper we introduce a construction of directed strongly regular graphs
from smaller ones using equitable partitions. Each equitable partition of a
single DSRG satisfying several conditions leads to an infinite family of
directed strongly regular graphs. We construct in this way dozens of infinite
families. For order at most 110, we confirm the existence of DSRGs for 30
previously open parameter sets.Comment: 27 page
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