Strongly regular graph

Remarks

Many authors also require that $G$ is neither complete nor edgeless, so that both conditions 2 and 3 apply to some pair. Counting the edges between the neighbours of a vertex $x$ and the vertices not adjacent to $x$ gives $k(k - \lambda - 1) = (v - k - 1)\mu$. Example: the Petersen graph is an $\mathrm{srg}(10, 3, 0, 1)$: it is $3$-regular, it has no triangles, so $\lambda = 0$, and two non-adjacent vertices have exactly one common neighbour. Check of the identity: $3 \cdot (3 - 0 - 1) = 6 = (10 - 3 - 1) \cdot 1$. Non-example: the cycle graph $C_6$ is $2$-regular, but two vertices at distance $2$ have one common neighbour while two opposite vertices have none, so clause 3 fails. The complement of an $\mathrm{srg}(v, k, \lambda, \mu)$ is an $\mathrm{srg}(v, v - k - 1, v - 2 - 2k + \mu, v - 2k + \lambda)$.