Seminár z teórie grafov - Soňa Pavlíková (11.4.2019)
vo štvrtok 11.4.2019 o 9:50 hod. v miestnosti M/213
Od: Martin Škoviera
Prednášajúci: Soňa Pavlíková (STU Bratislava)
Názov: Inverting non-invertible labeled trees
Termín: 11.4.2019, 9:50 hod., M/213
If a graph with non-zero edge labels has a non-singular adjacency matrix, then one may use the inverse matrix to define a (labeled) graph that may be considered to be the inverse graph to the original one. It has been known that an adjacency matrix of a labeled tree is non-singular if and only if the tree has a unique perfect matching. In the opposite case one may use a generalized inverse (which, in the symmetric case, coincides with Moore-Penrose, Drazin, or group inverse) of the adjacency matrix to `invert' a tree. A formula for entries of such a generalized inverse of a tree follows from the work of Britz, Olesky and van den Driessche (2004), based on a general formula for determining the Moore-Penrose inverse.
In our talk we will briefly introduce various approaches to `inverting' non-invertible matrices, state a formula for a generalized inverse of (an adjacency matrix of) a labeled tree, and outline principles leading to a new proof of validity of this formula (based solely on considering eigenvectors).