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Abstract. We seek solutions u ∈ R n to the semilinear elliptic partial difference equation −Lu + fs(u) = 0, where L is the matrix corresponding to the Laplacian operator on a graph G and fs is a one-parameter family of nonlinear functions. This article combines the ideas introduced by the authors in two papers: a) Nonlinear Elliptic Partial Difference Equations on Graphs (J. Experimental Mathematics, 2006), which introduces analytical and numerical techniques for solving such equations, and b) Symmetry and Automated Branch Following for a Semilinear Elliptic PDE on a Fractal Region (SIAM J. of Dynamical Systems, 2006), wherein we present some of our recent advances concerning symmetry, bifurcation, and automation for PDE.We apply the symmetry analysis found in the SIAM paper to arbitrary graphs in order to obtain better initial guesses for Newton's method, create informative graphics, and better understand the role of symmetry in the underlying variational structure. We use two modified implementations of the gradient Newton-Galerkin algorithm (GNGA, Neuberger and Swift) to follow bifurcation branches in a robust way. By handling difficulties that arise when encountering accidental degeneracies and higher-dimensional critical eigenspaces, we can find many solutions of many symmetry types to the discrete nonlinear system. We present a selection of experimental results which demonstrate our algorithm's capability to automatically generate bifurcation diagrams and solution graphics starting with only an edgelist of a graph. We highlight interesting symmetry and variational phenomena.

In this paper we numerically solve the eigenvalue problem ∆u+λu = 0 on the fractal region defined by the Koch Snowflake, with zero-Dirichlet or zero-Neumann boundary conditions. The Laplacian with boundary conditions is approximated by a large symmetric matrix. The eigenvalues and eigenvectors of this matrix are computed by ARPACK. We impose the boundary conditions in a way that gives improved accuracy over the previous computations of Lapidus, Neuberger, Renka & Griffith. We extrapolate the results for grid spacing h to the limit h → 0 in order to estimate eigenvalues of the Laplacian and compare our results to those of Lapdus et al. We analyze the symmetry of the region to explain the multiplicity-two eigenvalues, and present a canonical choice of the two eigenfunctions that span each two-dimensional eigenspace. √ 3 3 centered about the origin. With this choice, the polygonal approximations used 2000 Mathematics Subject Classification. 20C35, 35P10, 65N25.

We define a graph network to be a coupled cell network where there are only one type of cell and one type of symmetric coupling between the cells. For a difference-coupled vector field on a graph network system, all the cells have the same internal dynamics, and the coupling between cells is identical, symmetric, and depends only on the difference of the states of the interacting cells. We define four nested sets of difference-coupled vector fields by adding further restrictions on the internal dynamics and the coupling functions. These restrictions require that these functions preserve zero or are odd or linear. We characterize the synchrony and anti-synchrony subspaces with respect to these four subsets of admissible vector fields. Synchrony and anti-synchrony subspaces are determined by partitions and matched partitions of the cells that satisfy certain balance conditions. We compute the lattice of synchrony and anti-synchrony subspaces for some examples of graph networks. We also apply our theory to systems of coupled van der Pol oscillators.2000 Mathematics Subject Classification. 34C15, 34C23, 34A34, 37C80.

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