Symmetries of Hexagonal Molecular Graphs on the Torus

Symmetric properties of some molecular graphs on the torus are studied. In particular we determine which cubic cyclic Haar graphs are 1-regular, which is equivalent to saying that their line graphs are ½-arc-transitive. Although these symmetries make all vertices and all edges indistinguishable, they imply intrinsic chirality of the corresponding molecular graph

On a Cohen-Lenstra Heuristic for Jacobians of Random Graphs

In this paper, we make specific conjectures about the distribution of Jacobians of random graphs with their canonical duality pairings. Our conjectures are based on a Cohen-Lenstra type heuristic saying that a finite abelian group with duality pairing appears with frequency inversely proportional to the size of the group times the size of the group of automorphisms that preserve the pairing. We conjecture that the Jacobian of a random graph is cyclic with probability a little over .7935. We determine the values of several other statistics on Jacobians of random graphs that would follow from our conjectures. In support of the conjectures, we prove that random symmetric matrices over the p-adic integers, distributed according to Haar measure, have cokernels distributed according to the above heuristic. We also give experimental evidence in support of our conjectures.Comment: 20 pages. v2: Improved exposition and appended code used to generate experimental evidence after the \end{document} line in the source file. To appear in J. Algebraic Combi