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From what I've read I've gathered the following facts:

  • There are seven known such graphs.
  • Certain parameter sets are ruled out by the Krein conditions and the absolute bound.
  • Beyond that, little or nothing is known.

Am I missing something? I have read Biggs's report which lists all small feasible parameter sets and apparently this paper shows that (324,57,0,12) is infeasible.

Is something else known about this problem?

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The complete bipartite graphs $K_{n,n}$ are strongly regular and triangle-free. This nitpicking aside, your summary is accurate.

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    $\begingroup$ In fact, 20 years ago (almost) Chris published his "Problems in Algebraic Combinatorics" in a very early volume of the Electronic Journal of Combinatorics asking among other things, "Is there an eighth triangle-free SRG?" the answer to which is still unknown. $\endgroup$
    – Gordon Royle
    Commented Sep 11, 2013 at 8:55
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an update of Biggs's list (complete up to 1300 vertices) is maintained here by Andries Brouwer.

for a list of open problems and research directions, a good starting point could be Matan Ziv-Av's recent presentations 2011 --- 2013 and conference paper.

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