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 $((m+1)s,ls,ld,ld-d,ld)$ where $d,l$ and $s$ are integers with $dm=ls$ and $1\leq l<m$. We also give a new block matrix characterization for directed strongly regular graphs with parameters $(m(dm+1),dm,m,m-1,m)$, 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 $\mathcal{C}(G,S)$ is not a directed strongly regular graph if $S$ is a union of some conjugate classes of $G$.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 $\mathcal{C}(N\rtimes_\theta H, N_1\times H_1)$ with $N_1\subseteq N$ and $H_1\subseteq H$,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 $\mathcal{C}(D_n,X\cup Xa)$ with $X\cap X^{(-1)}=\emptyset$.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 $$(l(q-1)q^l,\ l(q-1)q^{l-1},\ (lq-l+1)q^{l-2},\ (l-1)(q-1)q^{l-2},\ (lq-l+1)q^{l-2})$$ for integers $q$ and $l$ ($q, l\ge 2$), and $$(lq^2(q-1),\ lq(q-1),\ lq-l+1,\ (l-1)(q-1),\ lq-l+1)$$ for all prime powers $q$ and $l\in \{1, 2,..., q\}$. The new graphs given by these constructions have parameters $(36, 12, 5, 2, 5)$, $(54, 18, 7, 4, 7)$, $(72, 24, 10, 4, 10)$, $(96, 24, 7, 3, 7)$, $(108, 36, 14, 8, 14)$ and $(108, 36, 15, 6, 15)$ listed as feasible parameters on "Parameters of directed strongly regular graphs," at ${http://homepages.cwi.nl/^\sim aeb/math/dsrg/dsrg.html}$ 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 $D_{p^\alpha}$,where $p$ is a prime and $\alpha\geqslant 1$ 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 $1$. 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 $F_n$ and $FP_n$ 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 $\langle A \mid r=1 \rangle$. In particular we show that all such monoids are of type $F_{\infty}$ (and $FP_{\infty}$), and that when $r$ is not a proper power, then the monoid has geometric and cohomological dimension at most $2$. 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