## Directed strongly regular graphs with rank 5

Preprint: Report R-2014-05.pdf

It is proved that a directed strongly regular with adjacency matrix of rank 5 has one the following parameter sets:

(where m is a positive integer)

- (20m,4m,m,0,m)
- (36m,12m,5m,2m,5m)
- (10m,4m,2m,m,2m)
- (16m,8m,5m,3m,5m)
- (20m,12m,9m,6m,9m)
- (18m,12m,10m,7m,10m)

or is the undirected strongly regular graph (5m,4m,4m,3m,4m).

For each of these families, a construction of directed strongly regular graphs is known for all values of m.

The result excludes existence of directed strongly regular graphs with the following previously open cases with n <=110.

(see Brouwer and Hobart: http://homepages.cwi.nl/~aeb/math/dsrg/dsrg.html )

- (45, 12, 4, 1, 4)
- (45, 24, 16, 10, 16)
- (49, 28, 20, 13, 20)
- (64, 16, 5, 1, 5)
- (80, 24, 9, 3, 9)
- (80, 56, 49, 35, 49)
- (81, 36, 20, 11, 20)
- (90, 24, 8, 2, 8)
- (90, 48, 32, 20, 32)
- (98, 28, 10, 3, 10)
- (98, 56, 40, 26, 40)