Speaker: Anna Bień (UŚ)

Title of the talk: Klasy grafów singularnych

Abstract: We consider simple graphs and their adjacency matrices. Graphs with singular adjacency matrix are called singular. Rara presented tools, which are useful in computing determinant of adjacency matrix of some simple graphs. Rara's methods allow to replace complicated algebraic calculations with operations performed on graphs. In some cases removing sets of edges or vertices does not change or changes the determinant of a graph in a specific way. We continue this subject matter and present general methods of reducing graphs. The most general is the method of identifying P₃ paths.
Consequences of this theorem are the method of contracting P₅ path and a method which can be applied to graphs circumscribed on cycles. We apply these methods in computing determinant of adjacency matrix of certain classes of graphs. In particular we present a recursive formula for planar grids P_n × P_n+1, det A(P_n × P_n+1) = (-1)^(n+1)/2 which is a main step in solution of the singularity problem for all planar grids.