On the homology groups of clique complexes of strongly regular graphs

Abstract

For a graph $G$ and a field $\mathbb{F},$ the first clique-homology $H_1(\text{Cl}(G),\mathbb{F})$ vanishes precisely when the cycle space of $G$ over $\mathbb{F}$ is generated by the signed boundaries of the triangles. We develop cycle-surgery methods for establishing this property in arbitrary characteristic and apply them to strongly regular graphs. Combining our results with Neumaier's classification, we show that $H_1(\text{Cl}(G),\mathbb{F})\ne 0$ can only occur in the Petersen graph, the Shrikhande graph, the complete bipartite graphs, the conference graphs on at most $255$ vertices, the lattice graphs, and the finite exceptional families $E_m$ in Neumaier's classification of strongly regular graphs with smallest adjacency eigenvalue $-m$, for some integer $m\ge 3$. Consequently, if $(G_n{)}_{n\ge 1}$ is an infinite family of pairwise distinct strongly regular graphs and $({\mathbb{F}}_n{)}_{n\ge 1}$ is a sequence of fields such that $H_1(\text{Cl}(G_n),{\mathbb{F}}_n)\ne 0$ for every $n$, then either $G_n$ is a lattice graph for infinitely many $n$, or ${\lim }_{n\to +\infty }{\lambda }_{\min }(G_n)=-\infty$. For Latin square graphs, we determine the clique homologies over arbitrary fields and show that if $G$ is the strongly regular graph associated with a Latin square $M$ of order $n\ge 5$ and $\mathbb{F}$ is any field, then $H_i(\text{Cl}(G),\mathbb{F})=0$ for $i=1$ or $i\ge 3$, and $\dim H_2(\text{Cl}(G),\mathbb{F})=(n-1{)}^3-I(M),$ where $I(M)$ is the number of $2\times 2$ Latin subsquares or intercalates in $M$.

40 pages, 4 figures