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We seek solutions u∈^n to the semilinear elliptic partial difference equation -Lu + f_s(u) = 0, where L is the matrix corresponding to the Laplacian operator on a graph G and f_s 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) SymmetryarXiv:1010.2257v1 fatcat:x3cv72ksejc6fj3beyxfu6s64a