  According to a well-know theorem by Sturm, a vibrating string is divided into
exactly N nodal intervals by zeros of its N-th eigenfunction. Courant showed
that one half of Sturm's theorem for the strings applies to the theory of
membranes: N-th eigenfunction cannot have more than N domains. He also gave an
example of a eigenfunction high in the spectrum with a minimal number of nodal
domains, thus excluding the existence of a non-trivial lower bound. An analogue
of Sturm's result for discretizations of the interval was discussed by
Gantmacher and Krein. The discretization of an interval is a graph of a simple
form, a chain-graph. But what can be said about more complicated graphs? It has
been known since the early 90s that the nodal count for a generic eigenfunction
of the Schrodinger operator on quantum trees (where each edge is identified
with an interval of the real line and some matching conditions are enforced on
the vertices) is exact too: zeros of the N-th eigenfunction divide the tree
into exactly N subtrees. We discuss two extensions of this result in two
directions. One deals with the same continuous Schrodinger operator but on
general graphs (i.e. non-trees) and another deals with discrete Schrodinger
operator on combinatorial graphs (both trees and non-trees). The result that we
derive applies to both types of graphs: the number of nodal domains of the N-th
eigenfunction is bounded below by N-L, where L is the number of links that
distinguish the graph from a tree (defined as the dimension of the cycle space
or the rank of the fundamental group of the graph). We also show that if it the
genericity condition is dropped, the nodal count can fall arbitrarily far below
the number of the corresponding eigenfunction.
