  In this paper we derive an analytical expression for the mean load at each
node of an arbitrary undirected graph for the uniform multicommodity flow
problem under random walk routing. We show the mean load is linearly dependent
on the nodal degree with a common multiplier equal to the sum of the inverses
of the non-zero eigenvalue of the graph Laplacian. Even though some aspects of
the mean load value, such as linear dependence on the nodal degree, are
intuitive and may be derived from the equilibrium distribution of the random
walk on the undirected graph, the exact expression for the mean load in terms
of the full spectrum of the graph has not been known before. Using the explicit
expression for the mean load, we give asymptotic estimates for the load on a
variety of graphs whose spectral density are well known. We conclude with
numerical computation of the mean load for other well-known graphs without
known spectral densities.
