Parameters of a strongly regular graph on 2k+3 vertices I'd like to find the parameters λ and μ (in terms of k) for a strongly regular graph G on 2k+3 vertices with valency k, where k>3. I'm a little stuck on how to go about finding these parameters. I have tried to consider the possible eigenvalues for the corresponding adjacency matrix and their multiplicities but have had no luck.
I have also noted that the smallest graph satisfying these properties has parameters (15,6,1,3) according to http://www.maths.gla.ac.uk/~es/srgraphs.php.
If anybody would be able to give some hints and/or pointers (rather than a full proof) as to how one could go about finding the last 2 parameters that would be much appreciated.
 A: If you substitute $v = 2k+3$ into the parameter relation for a strong regular graph srg$(v,k,\lambda,\mu)$, the result is a quadratic equation for $k$:
$$ k^2 - (\mu + \lambda + 1)k - 2\mu = 0 $$
From the quadratic formula:
$$ k = \frac{\mu + \lambda + 1 \pm \sqrt{(\mu + \lambda + 1)^2 + 8\mu}}{2} $$
To be valid graph parameters requires that $k$ is a positive integer for some non-negative integer values $\lambda,\mu$.  So the expression under the radical needs to be a perfect square, and we will choose only the plus sign (to avoid getting a spurious non-positive $k$).
[One also worries about division by two being exact, but as we will see this is guaranteed by the parity of $\mu + \lambda + 1$ matching the parity of the square root.]
Your example of srg$(15,6,1,3)$ illustrates what is required:
$$ k = \frac{5+7}{2} = 6 $$
To find additional possibilities we look for a difference of $8\mu$ between the perfect square $(\mu + \lambda + 1)^2$ and the one we want to appear under the square root radical, $(\mu + \lambda + 1)^2 + 8\mu$.  For each $\mu \gt 0$ there can be only finitely many such differences, as you can tell from the factors of a difference of two squares being (in this case) $8\mu$.
There is little more to say without trespassing into the "full proof" territory.  However note that if the difference between two squares is $8\mu$, that even difference implies the squares have the same even/odd parity, and thus their respective square roots will both be even or both odd.  This is our guarantee that the division by two in the quadratic formula gives an exact integer quotient.
