  The zero locus of a function f on a graph G is defined as the graph with
vertex set consisting of all complete subgraphs of G, on which f changes sign
and where x,y are connected if one is contained in the other. For d-graphs,
finite simple graphs for which every unit sphere is a d-sphere, the zero locus
of (f-c) is a (d-1)-graph for all c different from the range of f. If this Sard
lemma is inductively applied to an ordered list functions f_1,...,f_k in which
the functions are extended on the level surfaces, the set of critical values
(c_1,...,c_k) for which F-c=0 is not a (d-k)-graph is a finite set. This
discrete Sard result allows to construct explicit graphs triangulating a given
algebraic set. We also look at a second setup: for a function F from the vertex
set to R^k, we give conditions for which the simultaneous discrete algebraic
set { F=c } defined as the set of simplices of dimension in {k, k+1,...,n} on
which all f_i change sign, is a (d-k)-graph in the barycentric refinement of G.
This maximal rank condition is adapted from the continuum and the graph { F=c }
is a (n-k)-graph. While now, the critical values can have positive measure, we
are closer to calculus: for k=2 for example, extrema of functions f under a
constraint {g=c} happen at points, where the gradients of f and g are parallel
D f = L D g, the Lagrange equations on the discrete network. As for an
application, we illustrate eigenfunctions of geometric graphs and especially
the second eigenvector of 3-spheres, which by Courant-Fiedler has exactly two
nodal regions. The separating nodal surface of the second eigenfunction f_2
belonging to the smallest nonzero eigenvalue always appears to be a 2-sphere in
experiments if G is a 3-sphere.

Discrete Mathematics;Computational Geometry;Geometric Topology