Sporadic Examples of Directed Strongly Regular Graphs Obtained By Computer Algebra Experimentation

Abstract

We report about the results of the application of modern computer algebra tools for construction of directed strongly regular graphs. The suggested techniques are based on the investigation of non-commutative association schemes and Cayley graphs over non-Abelian groups. We demonstrate examples of directed strongly regular graphs for 28 different parameter sets, for which the existence of a corresponding digraph has not been known before.

Infinite families of directed strongly regular graphs using equitable partitions

“…This orbit-partition is good for the unique solution is proved for all positive integer j. For j = 1 the resulting parameter set (78, 36,23,16,17) is new according to [4]. (3,9).…”

“…Hence the existence of a DSRG with parameters (54j + 18, 18j + 4, 6j + 3, 6j, 6j + 1) is proven. For j = 1 a DSRG with parameter set (72, 22,9,6,7) has been recently found in [17]. 6.8 DSRG(18,7,5,2,3) For a DSRG (18,7,5,2,3) with adjacency matrix A 18,7 we have to consider two possibilities: (a, b) = (2,9) and (a, b) = (3,6).…”

“…This confirms the existence of DSRGs with parameters (52j +26, 26j +11, 13j + 7, 13j + 4, 13j + 5). For j = 1 the parameter set (78, 37,20,17,18) is new according to [4]. 6.10 SRG (26,10,3,4) According to Paulus [31] there are precisely 10 SRGs with parameters (26,10,3,4).…”

Directed Strongly Regular Cayley Graphs over Metacyclic Groups of Order 4n

“…Thus X satisfies the conditions (i) and (ii) of the Theorem 1. So we have the Cayley graph C(T 24 , a X ∪ a X b ∪ a X b 2 ∪ a X b 3 ) is a DSRCG with parameters (24, 8,4,0,4), where X = {1, 4}. Theorem 2.…”

“…Observe that the DSRGs have several parameters, there has been many constructions oriented to obtain several infinite families of DSRGs, also, some sporadic examples are known in the literature. Although many scholars have studied the existence and constructions of DSRGs for different parameters (one may refer to [2][3][4][5]), there are also plenty of DSRGs whose existence cannot be determined. As such, the complete characterization of DSRGs is far from being solved.…”