2026/06/25 by Sebastian M. Cioabă, Mutasim Mim
#math.CO
For a graph G and a field \mathbbF, the first clique-homology H1(Cl(G),\mathbbF) vanishes precisely when the cycle space of G over \mathbbF 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 H1(Cl(G),\mathbbF) ≠ 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 Em in Neumaier's classification of strongly regular graphs with smallest adjacency eigenvalue -m, for some integer m ≥ 3. Consequently, if (Gn)n≥ 1 is an infinite family of pairwise distinct strongly regular graphs and (\mathbbFn)n≥ 1 is a sequence of fields such that H1(Cl(Gn), \mathbbFn)\not=0 for every n, then either Gn is a lattice graph for infinitely many n, or limn→ +∞ λmin(Gn)=-∞. 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 ≥ 5 and \mathbbF is any field, then Hi(Cl(G),\mathbbF)=0 for i=1 or i ≥ 3, and dim H2(Cl(G),\mathbbF)=(n-1)3-I(M), where I(M) is the number of 2 × 2 Latin subsquares or intercalates in M.