  We give a polynomial-time algorithm for computing upper bounds on some of the
smaller energy eigenvalues in a spin-1/2 ferromagnetic Heisenberg model with
any graph $G$ for the underlying interactions. An important ingredient is the
connection between Heisenberg models and the symmetric products of $G$. Our
algorithms for computing upper bounds are based on generalized diameters of
graphs. Computing the upper bounds amounts to solving the minimum assignment
problem on $G$, which has well-known polynomial-time algorithms from the field
of combinatorial optimization. We also study the possibility of computing the
lower bounds on some of the smaller energy eigenvalues of Heisenberg models.
This amounts to estimating the isoperimetric inequalities of the symmetric
product of graphs. By using connections with discrete Sobolev inequalities, we
show that this can be performed by considering just the vertex-induced
subgraphs of $G$. If our conjecture for a polynomial time approximation
algorithm to solve the edge-isoperimetric problem holds, then our proposed
method of estimating the energy eigenvalues via approximating the
edge-isoperimetric properties of vertex-induced subgraphs will yield a
polynomial time algorithm for estimating the smaller energy eigenvalues of the
Heisenberg ferromagnet.
