Editing Strongly regular graph (section)

===The Hoffman–Singleton theorem===
As noted above, the multiplicities of the eigenvalues are given by 
:<math>M_{\pm} = \frac{1}{2}\left[(v - 1) \pm \frac{2k + (v - 1)(\lambda - \mu)}{\sqrt{(\lambda - \mu)^2 + 4(k - \mu)}}\right]</math>
which must be integers.

In 1960, [[Alan J. Hoffman|Alan Hoffman]] and Robert Singleton examined those expressions when applied on [[Moore graph]]s that have ''λ'' = 0 and ''μ'' = 1. Such graphs are free of triangles (otherwise ''λ'' would exceed zero) and quadrilaterals (otherwise ''μ'' would exceed 1), hence they have a girth (smallest cycle length) of 5. Substituting the values of ''λ'' and ''μ'' in the equation <math>(v - k - 1)\mu = k(k - \lambda - 1)</math>, it can be seen that <math>v = k^2 + 1</math>, and the eigenvalue multiplicities reduce to
:<math>M_{\pm} = \frac{1}{2}\left[k^2 \pm \frac{2k - k^2}{\sqrt{4k - 3}}\right]</math>

For the multiplicities to be integers, the quantity <math>\frac{2k - k^2}{\sqrt{4k - 3}}</math> must be rational, therefore either the numerator <math>2k - k^2</math> is zero or the denominator <math>\sqrt{4k - 3}</math> is an integer.

If the numerator <math>2k - k^2</math> is zero, the possibilities are:
* ''k'' = 0 and ''v'' = 1 yields a trivial graph with one vertex and no edges, and 
* ''k'' = 2 and ''v'' = 5 yields the 5-vertex [[cycle graph]] <math>C_5</math>, usually drawn as a [[regular pentagon]].

If the denominator <math>\sqrt{4k - 3}</math> is an integer ''t'', then <math>4k - 3</math> is a perfect square <math>t^2</math>, so <math>k = \frac{t^2 + 3}{4}</math>. Substituting:

:<math>\begin{align}
M_{\pm} &= \frac{1}{2} \left[\left(\frac{t^2 + 3}{4}\right)^2 \pm \frac{\frac{t^2 + 3}{2} - \left(\frac{t^2 + 3}{4}\right)^2}{t}\right] \\
32 M_{\pm} &= (t^2 + 3)^2 \pm \frac{8(t^2 + 3) - (t^2 + 3)^2}{t} \\
 &= t^4 + 6t^2 + 9 \pm \frac{- t^4 + 2t^2 + 15}{t} \\
 &= t^4 + 6t^2 + 9 \pm \left(-t^3 + 2t + \frac{15}{t}\right)
\end{align}</math>

Since both sides are integers, <math>\frac{15}{t}</math> must be an integer, therefore ''t'' is a factor of 15, namely <math>t \in \{\pm 1, \pm 3, \pm 5, \pm 15\}</math>, therefore <math>k \in \{1, 3, 7, 57\}</math>. In turn:
* ''k'' = 1 and ''v'' = 2 yields a trivial graph of two vertices joined by an edge,
* ''k'' = 3 and ''v'' = 10 yields the [[Petersen graph]],
* ''k'' = 7 and ''v'' = 50 yields the [[Hoffman–Singleton graph]], discovered by Hoffman and Singleton in the course of this analysis, and
* ''k'' = 57 and ''v'' = 3250 predicts a famous graph that has neither been discovered since 1960, nor has its existence been disproven.<ref>{{citation
 | last = Dalfó | first = C.
 | doi = 10.1016/j.laa.2018.12.035
 | journal = Linear Algebra and Its Applications
 | mr = 3901732
 | pages = 1–14
 | title = A survey on the missing Moore graph
 | volume = 569
 | year = 2019 | hdl = 2117/127212
 | s2cid = 126689579
 | hdl-access = free
}}</ref>

The Hoffman-Singleton theorem states that there are no strongly regular girth-5 Moore graphs except the ones listed above.