Mathematical investigation demonstrates the existence of directed strongly regular graphs on 54 and 108 vertices, indicating new symmetric network families in graph theory.
In this paper, we prove the existence of directed strongly regular graphs with parameters (108,11,3,2,1), (108,14,10,0,2), (108,22,12,6,4), (108,23,9,8,4), (108,25,15,8,5), (108,34,18,12,10), (108,38,22,12,14),(108,39,23,14,14), (108,41,35,16,15), (108,42,33,18,15) and (108,46,22,19,20). Further, we obtain directed strongly regular graphs with parameters (54,10,4,1,2), (54,11,4,3,2) and (54,16,7,4,5). The constructions are obtained by considering finite groups acting transitively on 54 and 108 vertices.
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Crnković et al. (2026) studied this question.
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