Editing Strongly regular graph (section)

===Eigenvalues and graph spectrum===
Since the adjacency matrix A is symmetric, it follows that [[orthogonal basis|its eigenvectors are orthogonal]]. We already observed one eigenvector above which is made of all ones, corresponding to the eigenvalue ''k''. Therefore the other eigenvectors ''x'' must all satisfy <math>Jx = 0</math> where ''J'' is the all-ones matrix as before. Take the previously established equation:
:<math>A^2 = kI + \lambda{A} + \mu(J - I - A)</math>
and multiply the above equation by eigenvector ''x'':
:<math>A^2 x = kIx + \lambda{A}x + \mu(J - I - A)x</math>
Call the corresponding eigenvalue ''p'' (not to be confused with <math>\lambda</math> the graph parameter) and substitute <math>Ax = px</math>, <math>Jx = 0</math> and <math>Ix = x</math>:
:<math>p^2 x = kx + \lambda p x - \mu x - \mu p x</math>
Eliminate x and rearrange to get a quadratic:
:<math>p^2 + (\mu - \lambda ) p - (k - \mu) = 0</math>

This gives the two additional eigenvalues <math>\frac{1}{2}\left[(\lambda - \mu) \pm \sqrt{(\lambda - \mu)^2 + 4(k - \mu)}\,\right]</math>. There are thus exactly three eigenvalues for a strongly regular matrix.

Conversely, a connected regular graph with only three eigenvalues is strongly regular.<ref>Godsil, Chris; Royle, Gordon. ''Algebraic Graph Theory''. Springer-Verlag, New York, 2001, Lemma 10.2.1.</ref>

Following the terminology in much of the strongly regular graph literature, the larger eigenvalue is called ''r'' with multiplicity ''f'' and the smaller one is called ''s'' with multiplicity ''g''.

Since the sum of all the eigenvalues is the [[Trace (linear algebra)|trace of the adjacency matrix]], which is zero in this case, the respective multiplicities ''f'' and ''g'' can be calculated:
* Eigenvalue ''k'' has [[Multiplicity (mathematics)|multiplicity]] 1.
* Eigenvalue <math>r = \frac{1}{2}\left[(\lambda - \mu) + \sqrt{(\lambda - \mu)^2 + 4(k - \mu)}\,\right]</math> has multiplicity <math>f = \frac{1}{2}\left[(v - 1) - \frac{2k + (v - 1)(\lambda - \mu)}{\sqrt{(\lambda - \mu)^2 + 4(k - \mu)}}\right]</math>
* Eigenvalue <math>s = \frac{1}{2}\left[(\lambda - \mu) - \sqrt{(\lambda - \mu)^2 + 4(k-\mu)}\,\right]</math> has multiplicity <math>g = \frac{1}{2}\left[(v - 1) + \frac{2k + (v - 1)(\lambda - \mu)}{\sqrt{(\lambda - \mu)^2 + 4(k - \mu)}}\right]</math>

As the multiplicities must be integers, their expressions provide further constraints on the values of ''v'', ''k'', ''μ'', and ''λ''.

Strongly regular graphs for which <math>2k + (v - 1)(\lambda - \mu) \ne 0</math> have integer eigenvalues with unequal multiplicities.

Strongly regular graphs for which <math>2k + (v - 1)(\lambda - \mu) = 0</math> are called [[conference graph]]s because of their connection with symmetric [[conference matrix|conference matrices]]. Their parameters reduce to
: <math>\operatorname{srg}\left(v, \frac{1}{2}(v - 1), \frac{1}{4}(v - 5), \frac{1}{4}(v - 1)\right).</math>
Their eigenvalues are <math>r =\frac{-1 + \sqrt{v}}{2}</math> and <math>s = \frac{-1 - \sqrt{v}}{2}</math>, both of whose multiplicities are equal to <math>\frac{v-1}{2}</math>. Further, in this case, ''v'' must equal the sum of two squares, related to the [[Bruck–Ryser–Chowla theorem]].

Further properties of the eigenvalues and their multiplicities are:<ref>Brouwer & van Meldeghem, ibid.</ref>
* <math>(A - rI)\times(A - sI) = \mu.J</math>, therefore <math>(k - r).(k - s) = \mu v</math>
* <math>\lambda - \mu = r + s</math>
* <math>k - \mu = -r\times s</math>
* <math>k \ge r</math>
* Given an {{nowrap|srg(''v'', ''k'', λ, μ)}} with eigenvalues ''r'' and ''s'', its complement {{nowrap|srg(''v'', ''v'' − ''k'' − 1, ''v'' − 2 − 2''k'' + μ, ''v'' − 2''k'' + λ)}} has eigenvalues ''-1-s'' and ''-1-r''.
* Alternate equations for the multiplicities are <math>f =\frac{(s+1)k(k-s)}{\mu(s-r)}</math> and <math>g =\frac{(r+1)k(k-r)}{\mu(r-s)}</math>
* The frame quotient condition: <math>v k (v-k-1) = f g (r-s)^2</math>. As a corollary, <math>v = (r-s)^2</math> [[if and only if]] <math>{f,g} = {k, v-k-1}</math> in some order.
* Krein conditions: <math>(v-k-1)^2 (k^2 + r^3) \ge (r+1)^3 k^2</math> and <math>(v-k-1)^2 (k^2 + s^3) \ge (s+1)^3 k^2</math>
* Absolute bound: <math>v \le \frac{f(f+3)}{2}</math> and <math>v \le \frac{g(g+3)}{2}</math>.
* Claw bound: if <math>r + 1 > \frac{s(s+1)(\mu+1)}{2}</math>, then <math>\mu = s^2</math> or <math>\mu = s(s+1)</math>.
If any of the above conditions are violated for a set of parameters, then there exists no strongly regular graph for those parameters. Brouwer has compiled such lists of existence or non-existence [https://www.win.tue.nl/~aeb/graphs/srg/srgtab.html here] with reasons for non-existence if any. For example, there exists no srg(28,9,0,4) because that violates one of the Krein conditions and one of the absolute bound conditions.