  We define the Euler number of a bipartite graph on $n$ vertices to be the
number of labelings of the vertices with $1,2,...,n$ such that the vertices
alternate in being local maxima and local minima. We reformulate the problem of
computing the Euler number of certain subgraphs of the Cartesian product of a
graph $G$ with the path $P_m$ in terms of self adjoint operators. The
asymptotic expansion of the Euler number is given in terms of the eigenvalues
of the associated operator. For two classes of graphs, the comb graphs and the
Cartesian product $P_2 \Box P_m$, we numerically solve the eigenvalue problem.
