Discrete mathematics seminar - On some optimisation problems for homogeneous algebraic graphs

Dates

Wednesday, March 22, 2023 - 15:00 to 16:00

Speaker: Vasyl Ustimenko (Royal Holloway)

Abstract:
Homogeneous algebraic graphs defined over an arbitrary field are classical objects of Algebraic Geometry. This class includes geometries of Chevalley groups A_2(F), B_2(F) and G_2(F) defined over an arbitrary field F. Assume that codimension of homogeneous graph is the ratio of dimension of variety of its vertices and the dimension of neighbourhood of some vertex. We evaluate the minimal codimension v(g) and u(h) of an algebraic graph of prescribed girth g and cycle indicator h. Recall that girth is the size of a minimal cycle in the graph and girth indicator stands for the maximal value of the shortest path through some vertex. We prove that for even h the inequality u(h)<=(h-2)/2 holds. We define a class of homogeneous algebraic graphs with even cycle indicator h and codimension (h-2)/2. It contains geometries A_2(F), B_2(F) and G_2(F) and infinitely many other homogeneous algebraic graphs.