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I am looking for an algorithm to determine whether a graph is Strongly Regular Graph (SRG) or not.

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  • $\begingroup$ There is no need for your first step, which also seems a bit too much to ask (how do you determine $\lambda,\mu$ from a single equation?). $\endgroup$ – Yuval Filmus Apr 28 '15 at 1:34
  • $\begingroup$ What have you tried? Where did you get stuck? What research have you done? We expect you to do a significant amount of research before asking, and to show us in the question what you've tried and what research you've done. $\endgroup$ – D.W. Apr 30 '15 at 22:45
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There is a trivial $O(kn^2)$ algorithm:

  1. Choose a vertex $v$ and calculate its degree $k$. Then verify that all other vertices have degree $k$.

  2. Choose some neighbor $w$ of $v$ and calculate the number of common neighbors $\lambda$. Then verify that all other pairs of adjacent vertices have $\lambda$ common neighbors.

  3. Choose some non-neighbor $x$ of $v$ and calculate the number of common neigbors $\mu$. Then verify that all other pairs of non-adjacent vertices have $\mu$ common neighbors.

If you like linear algebra, you can also compute the square $A^2$ of the adjacency matrix $A$. Then check that:

  1. All diagonal entries of $A^2$ are equal to some constant $k$.

  2. All off-diagonal entries $A^2_{ij}$ such that $A_{ij} = 1$ are equal to some constant $\lambda.

  3. All off-diagonal entries $A^2_{ij}$ such that $A_{ij} = 0$ are equal to some constant $\mu$.

This algorithm takes time $O(n^\omega)$.

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