# The number of non-isomorphic strongly regular graphs on n vertices

What is the number of strongly regular graphs on $n$ vertices? or at least how many non-isomorphic strongly regular graphs can exist?

• Thank you very much Professor Chris Godsil and professor Aaron Meyerowitz. – Mojtaba Jazaeri Feb 5 '13 at 17:54

## 1 Answer

No formula is known. Since Latin squares and Steiner triple systems give strongly regular graphs, lower bounds on the numbers of these structure give lower bounds if $n$ is square or if $n=v(v-1)/6$ and $v\equiv1,3$ mod 6. (Presumably these lower bounds are very weak.) According to Brouwer's tables we have exact enumeration up to 36; the numbers on 37 and 41 are not known. (I expect that the number of srgs on $p$ vertices, $p$ prime increases with $p$, but this has not been proved.)

Of course we do not know the number of isomorphism classes of graphs on $n$ vertices. We have a procedure that allows us to compute the number for moderate values of $n$, and we know that asymptotically the number is $2^{n(n-1)/2}/n!$. Given this, our knowledge for strongly regular graphs does not seem quite so bad.

• It is worth mentioning that the lower bounds in some case, while perhaps weak, are still enormous. The number of $m \times m$ latin squares is known to be greater than $\frac{m!^{2m}}{m^{m^2}}$ so I suppose this would be for $n=m^2$ and there is the mater of dividing out $m!^3$ to account for isomorphism/isotopys. But that is still a big number and accounts for both $n$ and the degree. Pairs, triples, etc. of pairwise orthogonal squares give more possibilities. – Aaron Meyerowitz Feb 5 '13 at 16:59
• It's true that the lower bounds we get from (say) Latin squares are enormous, but they're probably a really long way from the truth. – Chris Godsil Feb 5 '13 at 18:30
• As we know, every strongly regular graph over prime number of vertices is a conference graph and Paley graph is a conference graph and the following sentence due to Willem H. HAEMERS: For v = 5, 9, 13 and 17, the Paley graph is the only one with the given parameters. If $v \geq 25$, other graphs with the same parameters exist. in the following paper: Matrices for graphs, designs and codes Why this sentence is true? if this sentence is true, then there are at least 2 non-isomorphic strongly regular graphs on prime number of vertices $p>25$. – Mojtaba Jazaeri Feb 9 '13 at 18:52
• Ted Spence determined all conference graphs on 29 vertices (see maths.gla.ac.uk/~es/srgraphs.php), there were 41 of them. I have not seen a proof that for any prime $p$ with $p\ge29$, there are at least two conference graphs on $p$ vertices. If you want to know what Haemers meant, you will have to ask him. – Chris Godsil Feb 9 '13 at 19:59