Semisymmetric graphs defined by finite-dimensional generalized Kac-Moody algebras
This paper provides several results about the symmetries of graphs defined by Kac-Moody algebras. Broadly speaking, it classifies algebraically-defined graphs as bipartite graphs which are the incidence graphs of structures that consist of points and lines, with algebraic equations determining incidence relations. The paper focuses on two families of algebraically-defined graphs where the incidence relations come from Kac-Moody algebras. For one of these families, the authors find an abelian group that acts regularly on the points (and also on the lines) of the structure and an involution that interchanges the points with the lines and inverts the elements of the abelian group. They thereby conclude that these graphs are Cayley graphs on groups of dihedral type (I would use the terminology generalised dihedral groups). For completeness, their results also include a third family (the second listed in their results) but they say nothing new about this family, which was already known not to be vertex-transitive. For the last family, they show that there are structural differences in the graph between the set of points and the set of lines, so these graphs are not vertex-transitive. Since these graphs were already known to be edge-transitive, this means they are semisymmetric. The paper concludes with short proof that the first family is also Hamiltonian.
scientific article; zbMATH DE number 7610172
scientific article