Triangle free strongly regular graphs

Definitions and mathematical statement

A finite simple graph is strongly regular with parameters $(v,k,\lambda,\mu)$ if it has $v$ vertices, every vertex has degree $k$, every adjacent pair has exactly $\lambda$ common neighbors, and every distinct nonadjacent pair has exactly $\mu$ common neighbors. Require $0<k<v-1$. Call such a graph primitive when both the graph and its complement are connected. The triangle-free condition is $\lambda=0$.

The problem asks whether there exists a primitive triangle-free strongly regular graph outside the following seven isomorphism types, recorded together with their standard parameters: \[ \begin{array}{c|c} C_5&(5,2,0,1)\\ \text{Petersen}&(10,3,0,1)\\ \text{Clebsch}&(16,5,0,2)\\ \text{Hoffman--Singleton}&(50,7,0,1)\\ \text{Gewirtz}&(56,10,0,2)\\ M_{22}&(77,16,0,4)\\ \text{Higman--Sims}&(100,22,0,6). \end{array} \] These parameter sets identify the familiar seven examples. The question is about existence of an actual finite graph satisfying all common-neighbor conditions, not merely integer parameters satisfying the associated counting equations or eigenvalue restrictions. A negative answer would prove that this list is exhaustive up to graph isomorphism.

Primitivity is essential: complete bipartite graphs with equally sized parts are triangle-free strongly regular and would otherwise supply an immediate infinite family. The source discussion explicitly excludes that family. Disconnected unions of cliques are likewise removed by the connectivity convention. No upper bound is imposed on the order, degree, or number of common neighbors in a prospective additional example.

Short English statement

Is there an eighth primitive triangle-free strongly regular graph beyond the seven standard examples?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3145 (OPG-345), OpenGarden. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Triangle free strongly regular graphs. Original problem and discussion. http://www.openproblemgarden.org/op/triangle_free_strongly_regular_graphs.
• S3 C. Godsil, Geometry: algebraic graph theory notes, strongly regular graphs. https://www.math.uwaterloo.ca/~cgodsil/pdfs/bigGeom.pdf.