On the critical group of the missing Moore graph

@article{Ducey2017OnTC,
  title={On the critical group of the missing Moore graph},
  author={Joshua E. Ducey},
  journal={Discret. Math.},
  year={2017},
  volume={340},
  pages={1104-1109}
}
  • Joshua E. Ducey
  • Published 1 September 2015
  • Mathematics, Computer Science
  • Discret. Math.
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 group up to two possibilities. 

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