Activity: Talk or presentation › Science to science
Description
A method for characterizing graphs that have smallest (or largest)
Laplacian eigenvalue within a particular class of graphs works as
following: Take a Perron vector, rearrange the edges of the graph
and compare the respective Rayleigh quotients. By the Rayleigh-Ritz
Theorem we can draw some conclusions about the change of the smallest eigenvalue. This approach, however, does not work for the k-th Laplacian eigenvalue, as now we have to use the
Courant-Fisher Theorem that involves minimization of the Rayleigh quotients with respect to constraints that are hard to control.
In this talk we show that sometimes we can get local properties of extremal graphs by means of the concept of geometric nodal domains
and Dirichlet matrices. This is in particular the case for the algebraic connectivity.