Seminar Details

We classify the strongly regular graphs for which the first clique homology group can fail to vanish over some field. Using Neumaier's classification of strongly regular graphs with fixed smallest integral eigenvalue, together with vanishing results for the geometric and pseudogeometric families of graphs appearing there, we prove that nonvanishing $\hlcl{G}{\FF}$ can occur only in a short explicit collection of cases: the surviving graphs in Seidel's classification of strongly regular graphs with smallest adjacency eigenvalue $-2,$ complete bipartite graphs, graphs associated with the orthogonal arrays $OA(3,2),\ OA(3,3),\ OA(4,3)$, conference graphs on at most $255$ vertices, and the exceptional families $E_m$ for some $m \geq 3$ in Neumaier's classification. In particular, if $(G_n)_{n\geq 1}$ is an infinite family of pairwise distinct strongly regular graphs with $\hlcl{G}{\FF_n}\not=0$ for every $n$, then either $G_n$ is a lattice graph for infinitely many $n$, or $\lim_{n\rightarrow +\infty} \lambda_{\min}(G_n)=-\infty.$ Thus, the lattice graphs are the only surviving infinite family with bounded smallest eigenvalue. For Latin square graphs, we compute the remaining clique homologies completely: if $G$ is the strongly regular graph associated with a Latin square $M$ of order $n \geq 5$ and $\FF$ any field, then $H_i(\clc{G},\FF)=0$ for $i=1$ or $i \geq 3$, and $\dim H_2(\clc{G},\FF)=(n-1)^3-I(M),$ where $I(M)$ is the number of $2 \times 2$ sub-Latin squares in $M$.