  Given a length function on the edge set of a finite graph, we define a
vertex-weight and an edge-weight in terms of it and consider the corresponding
graph Laplacian. In this paper, we consider the problem of maximizing the first
nonzero eigenvalue of this Laplacian over all edge-length functions subject to
a certain normalization. For an extremal solution of this problem, we prove
that there exists a map from the vertex set to a Euclidean space consisting of
first eigenfunctions of the corresponding Laplacian so that the length function
can be explicitly expressed in terms of the map and the Euclidean distance.
This is a graph-analogue of Nadirashvili's result related to first-eigenvalue
maximization problem on a smooth surface. We discuss simple examples and also
prove a similar result for a maximizing solution of the
Goering-Helmberg-Wappler problem.
