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We seek solutions $u\in\R^n$ to the semilinear elliptic partial differenceequation $-Lu + f_s(u) = 0$, where $L$ is the matrix corresponding to theLaplacian operator on a graph $G$ and $f_s$ is a one-parameter family ofnonlinear functions. This article combines the ideas introduced by the authorsin two papers: a) {\it Nonlinear Elliptic Partial Difference Equations onGraphs} (J. Experimental Mathematics, 2006), which introduces analytical andnumerical techniques for solving such equations, and b) {\it Symmetry andAutomated Branch Following for a Semilinear Elliptic PDE on a Fractal Region}wherein we present some of our recent advances concerning symmetry,bifurcation, and automation fo  We apply the symmetry analysis found in the SIAM paper to arbitrary graphs inorder to obtain better initial guesses for Newton's method, create informativegraphics, and be in the underlying variational structure. We use two modifiedimplementations of the gradient Newton-Galerkin algorithm (GNGA, Neuberger andSwift) to follow bifurcation branches in a robust way. By handling difficultiesthat arise when encountering accidental degeneracies and higher-dimension wecan find many solutions of many symmetry types to the discrete nonlinearsystem. We present a selection of experimental results which demonstrate ouralgorithm's capability to automatically generate bifurcation diagrams andsolution graphics starting with only an edgelis of a graph. We highlightinteresting symmetry and variational phenomena.

Automated Bifurcation Analysis for Nonlinear Elliptic Partial Difference Equations on Graphs