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.