School of Mathematical Sciences

Local nodal surplus and nodal count distribution for graphs with disjoint loops. menu

Local nodal surplus and nodal count distribution for graphs with disjoint loops.

Speaker: 
Gregory Berkoliako (Texas A&M)
Date/Time: 
Tue, 13/06/2017 - 16:00
Room: 
W316
Seminar series: 

The nodal surplus of the $n$-th eigenfunction of a graph is defined as
the number of its zeros minus $(n-1)$. When the graph is composed of
two or more blocks separated by bridges, we propose a way to define a
"local nodal surplus" of a given block. Since the eigenfunction index
$n$ has no local meaning, the local nodal surplus has to be defined in
an indirect way via the nodal-magnetic theorem of Berkolaiko and
Weyand.

We will discuss the properties of the local nodal surplus and their
consequences. In particular, it also has a dynamical interpretation
as the number of zeros created inside the block (as opposed to those
who entered it from outside) and its symmetry properties allow us to
prove the long-standing conjecture that the nodal surplus distribution
for graphs with $\beta$ disjoint loops is binomial with parameters
$(\beta, 1/2)$. The talk is based on a work in progress with Lior Alon
and Ram Band.