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We consider the critical group of a hypothetical Moore graph of diameter 2 and valency 57 . Determining this group is equivalent to finding the Smith normal form of the Laplacian matrix of such a graph. We show that all of the Sylow p -subgroups of the critical group must be elementary abelian with the exception of p = 5 . We prove that the 5 -rank of the Laplacian matrix determines the critical...
In this paper, we study 24 possibilities of directed strongly regular graphs with adjacency matrix of rank 6, our proof is based on two effective algorithms. At last, we get 4 families of directed strongly regular graphs with realizable parameter sets, exclude 20 families of directed strongly regular graphs with feasible parameter sets; partially solve the classification problem of directed strongly...
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