Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1510.00110

Notes on Various Methods for Constructing Directed Strongly Regular Graphs

Jerod Michel·Baokun Ding
View original source ↗Read source document ↗Export metadata ↓

Abstract

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].

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jerod Michel, Baokun Ding. 2015-10-01. Notes on Various Methods for Constructing Directed Strongly Regular Graphs. https://arxiv.org/abs/1510.00110

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On nut graphs with two vertex and three edge orbits

Nut graphs are graphs whose adjacency matrix is singular with one-dimensional null space spanned by a vector with no zero entries. In a recent paper, Bašić, Fowler and Pisanski proved that the automorphism group of a nut graph has more orbits on the edge set than on the vertex set. They classified all orders for which a vertex-transitive nut graph with precisely two edge orbits exists, and conjectured that a nut graph with two vertex and three edge orbits exists for each non-prime order $n \ge 9$. Motivated by this conjecture, we introduce a very general construction that provides graphs with the desired symmetry properties, and we determine some sufficient spectral and structural conditions under which they are nut graphs. The construction yields infinite families of examples and confirms the above conjecture for all odd non-prime orders up to $2\,500$ and for at least $99.8$ percent of all odd non-prime orders up to a million. Finally, we present some additional interesting examples of nut graphs with two vertex and three edge orbits that do not arise from this construction.

math.CO↗
arXiv2026

On vertex-minimal simplicial maps to the sphere

For positive integers $n,d$, let $λ(n,d)$ be the minimal number of vertices of a triangulation of the $n$-sphere which admits a degree $d$ simplicial map onto the boundary of the $(n+1)$-simplex. We show that for $h=\lfloor\frac{n+1}2\rfloor$, the function $λ(n,d)^h$ has linear order of growth in $d$, answering a question of O. Musin. All triangulations we obtained are isomorphic to boundaries of convex polytopes in $\mathbb{R}^{n+1}$.

math.CO↗