New tools for the construction of directed strongly regular graphs: Difference digraphs and partial sum families

Abstract

We define a class of digraphs involving differences in a group that generalizes Cayley digraphs, that we call difference digraphs. We define also a new combinatorial structure, called partial sum family, or PSF for short, from which we obtain difference digraphs that are directed strongly regular graphs. We give an infinite family of PSFs and we give also twelve sporadic ones that generate directed strongly regular graphs whose existence was previously undecided.

Partial sum quadruples and bi-Abelian digraphs

“…With respect to family (iv), as we said after the proof of Theorem 4.11, besides of the known examples when U = 0 and s = 1, 2, 3, 4, 5 (see [27] and [22]) that correspond with the undirected case, examples generating DSRGs with new parameters were also found by the first and fourth author in [24] for s = 1, U = 5, s = 1, U = 6 and s = 2, U = 2.…”

“…Putting r = 2 f + 1 we obtain parameters in the form indicated in part (iii) of this theorem. 2 Proposition 3.7 with e = 2 shows that circulant PSQs with parameters as in part (ii) of the previous theorem always exists for all possible values of r and s, and Proposition 3.8 shows that also circulant PSQs with parameters as in part (iii) always exists for all possible values of r and s. With respect to PSQs with parameters as in part (i), apart from the small number of known examples when U = 0 (that is, when they originate undirected strongly regular graphs), in [24] examples were obtained for s = 1, U = 5, for s = 1, U = 6 and for s = 2, U = 2. Proof.…”

“…The strong regularity question for Cayley and bi-Cayley graphs is related to the notions of partial difference sets [4,20] and partial difference triples [19,27], whereas the strong regularity question for m-Cayley digraphs is related to the notion of partial sum families, first introduced in [24] in the context of the so-called difference digraphs which are defined as follows. Let G be a finite group, H a normal subgroup of G and ϕ : G/H × G/H → P(G − {e}) a mapping satisfying ϕ(x, y) ⊆ y − x for every x, y ∈ G/H.…”

“…Numerous papers have been published on the subject of DSRGs (see, for example, [5,6,8,[11][12][13][14]16,24]). In 1997 Klin, Munemasa, Muzychuk and Zieschang [15] proved that a DSRG cannot be a Cayley digraph of an abelian group, whereas in 1999, Hobart and Shaw [11] gave constructions of DSRGs which are Cayley digraphs of nonabelian groups.…”

QUASI m-CAYLEY STRONGLY REGULAR GRAPHS

“…Then SRGs admitting semiregular groups of automorphisms with three orbits were studied by Kutnar, Marušič, Miklavič andŠparl [6]. Recently, Martínez and Araluze [11] studied this question for an arbitrary number of orbits and for directed SRGs, introduced by Duval in [5]. They translated the problem into the language of the so-called partial sum families (see also [1,2]).…”