Faculty of Mathematics, Physics
and Informatics
Comenius University Bratislava

Seminar of Graph Theory- Soňa Pavlíková (20.10.2016)

Thursday 20.10.2016 at 9:50, Lecture room M/213

18. 10. 2016 09.24 hod.
By: Martin Škoviera

Soňa Pavlíková (STU Bratislava):
Inverses of graphs

In the first part of the talk we deal with simple labeled graphs with non-zero labels in a ring. If the adjacency matrix of such a graph is invertible, its inverse is an adjacency matrix of another graph, the inverse of the original graph. If the ring is ordered, then balanced inverses - those with a positive product of labels along every cycle - are of interest. We introduce the concept of a derived labeled graph and show how it can be embedded into an inverse. We also prove a new result on balanced inverses of labeled trees and present a construction of new labeled graphs with balanced inverses from old.

In the second part we investigate the so-called positively and negatively invertible graphs. The class of negatively invertible graphs turns out to contain models of important organic molecules and invertibility allows to derive bounds on the binding energy of such molecules. We will present a fairly general construction of new invertible graphs based on `bridging' a pair of invertible graphs, which, informally, means `joining' the pair by a new bipartite graph attached to the two graphs at suitable subsets of vertices.

This is a joint work with Daniel Sevcovic.