24 November 2017

ZiF Final Conference

Getting ready for the ZiF final conference in Bielefeld. I will talk about the nodal domains of the $p$-Laplacian operator on discrete graphs.

Precisely, this is the abstract of my talk: The number of nodal domains induced by the eigenfunctions of the Laplacian operator has been completely described both for graphs and for continuous domains. For $p\geq 1$, the $p$-Laplacian is a nonlinear operator which reduces to the standard Laplacian when $p=2$. This nonlinear operator has gained popularity in recent years as, for instance, it can be used to improve data clustering algorithms. We consider a set of variational eigenvalues of the $p$-Laplacian on discrete graphs and analyze the nodal domain structure of the associated eigenfunctions. We show that when $p>1$, the upper bound in the linear nodal domain theorem carries over unchanged to the nonlinear setting, whereas some properties are lost when $p=1$. We also discuss an higher-order Cheeger inequality that can be obtained by exploiting the nodal structure of the $p$-Laplacian.