This volume presents the proceedings of the workshop "Harmonic Functions on Graphs" held at the Graduate Center of CUNY in the fall of 1995. The main papers present material from four minicourses given by leading experts: D. Cartwright, A. Fig-Talamanca, S. Sawyer and T. Steger. These minicourses are introductions which gradually progress to deeper and less known branches of the subject. One of the topics treated is buildings, which are discrete analogues of symmetric spaces of arbitrary rank; buildings of rank are trees. Harmonic analysis on buildings is a fairly new and important field of research. One of the minicourses discusses buildings from the combinatorial perspective and another examines them from the $p$-adic perspective. The third minicourse deals with the connections of trees with $p$-adic analysis. And the fourth deals with random walks, i.e., with the probabilistic side of harmonic functions on trees. The book also contains the extended abstracts of 19 of the 20 lectures given by the participants on their recent results. These abstracts, well detailed and clearly understandable, give a good cross-section of the present state of research in the field.

Part 1. Minicourses 
A. Fig-Talamanca -- Local fields and trees 
S. A. Sawyer -- Martin boundaries and random walks 
D. I. Cartwright -- A brief introduction to buildings 
T. Steger -- Local fields and buildings 
Part II. Abstracts of Lectures 
D. Bednarchak -- Heat kernel for regular trees 
D. I. Cartwright, M. G. Kuhn, and P. M. Soardi -- Tensor product of spherical representations of the group of automorphisms of a homogeneous tree 
E. C. Tarabusi -- The horocyclic Radon transform on trees 
J. M. Cohen and F. Colonna -- Eigenfunctions of the Laplacian on a homogeneous tree 
F. Di Biase -- Exotic convergence in theorems of Fatou type 
Y. Guivarcl'h -- A spectral gap property for transfer operators 
V. A. Kaimanovich -- Harmonic functions on discrete subgroups of semi-simple Lie groups 
R. Lyons -- Biased random walks and harmonic functions on the Lamplighter group 
A. M. Mantero and A. Zappa -- Characterization of the eigenfunctions of the Laplace operators for an affine building of rank 2 
W. Mlotkowski -- Free product of representations 
T. Nagnibeda -- The Jacobian of a finite graph 
S. Northshield -- Flows and harmonic functions on graphs 
M. Pagliacci -- Applications of diffusion processes on trees to mathematical finance 
M. A. Picardello -- Characterizing harmonic functions by mean value properties on trees and symmetric spaces 
J. Ramagge and G. Robertson -- Factors from buildings 
M. Rigoli, M. Salvatori, and M. Vignati -- Harnack and Liouville properties on graphs 
G. Robertson -- The spectrum of a directed Cayley graph of a free group 
M. H. Taibleson -- Factorization of the Green's kernel for non-nearest neighbor random walks 
W. Woess -- Harmonic functions for group-invariant random walks

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