Updates and Questions about the Poincare-Hopf paper

Typos corrected (last: Jan 23, 2012): Remarks:
Question Answer
I am interested in your paper on the graph theoretical Poincare'-Hopf theorem but I have trouble understanding the unit sphere and gradient. The gradient at x is a function on the set S(x) of all points y which are connected to x. This is called the unit sphere of x. Let me draw an example of a graph, where the vertices are labeled with the values of the function f:
                 3      4
                  \     /\
                   \   /  \
               5-----2-----1
                \    |
                 \   |
                  \  |
                    6
The vertex 2 has 5 neighbors. This is the unit sphere of this vertex. and it is a graph too. The gradient is a function on the vertices y of this graph. The values are the difference f(y)-f(x). In the example:
         at 5:     5-2=3,
         at 6:     6-2=3,
         at 1:     1-2=-1,
         at 4:     4-2=2,
         at 3:     3-2=1,
The index is defined as 1 minus the Euler characteristic of the set S- where the gradient is negative. You see that S- = { 1 } consists of only one point. The index is 1-1 = 0. If we do the index computation at every point, then you get the following values:
                 0      0
                  \     /\
                   \   /  \
               0-----0-----1
                \    |
                 \   |
                  \  |
                    0
Only the minimum had nonzero index for this graph. The minimum often has index 1, especially in geometric situations. The sum of the indices is 1. Now lets compute the Euler characteristic. There are v=6 vertices, e=7 edges and f=2 triangles and no K4 graphs. The Euler characteristic is v-e+f= 6-7+2 = 1. The theorem assures that this agrees with the index sum. Here is an other example.