So I am currently working on the Hoffman singleton conjecture. For those who do not know this conjecture, it is asking whether there exists a 57-regular, girth 5 graph with $57^2+1$ vertices.

While working on it, I made some interesting observations:

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> Observation 1: The Peterson graph and the Hoffman-Singleton graph is made up of vertex disjoint copies of C5s.
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So is it possible to prove/disprove that a potential 57-regular girth 5 graph with $57^2+1$ vertices (if it exists) is also made up of vertex disjoint copies of the C5?

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> Observation 2: The Peterson graph is made up of 2 vertex disjoint copies of the C5 and that the Hoffman-Singleton graph is made up of 5 vertex disjoint copies of the Peterson graph.
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So can we prove/disprove that a potential 57-regular, girth 5 graph with $57^2+1=3250$ vertices (if it exists) is made up of $\frac{3250}{50}=65$ vertex disjoint copies of the Hoffman-Singleton graph?

Things that I have tried: The Cauchy Interlace theorem.

I am open to any ideas that you have tried, regardless of whether you succeeded or not.