Typos corrected (last: Jan 23, 2012):
• Page 7: let W_k be the number of K_{k+1} (instead of W_i be the number .....)
• Page 8: V_k(v) - W_k(v) instead of V_k-(v) W_k(v) and leave i(A(v)) = ... away. A is not a graph since no vertices. This simplicial complex will appear in future work.
• Figure 6 and 9 have identical pictures (placing them into one directory for the arxiv trashed one).
Remarks:
• We have seen that the index of a gradient field dimensional graph G takes values less or equal to 1 as for gradient fields on two dimensional manifolds. Cases like vector fields on the two sphere with one equilibrium point of index 2 are not gradient fields. The remark on indices of gradient fields was in mentioned in Arnold's book on mechanics in Appendix 9 (page 419 in the English edition). (I learned this reference from "Pete Gabor, Morse Theory, 20 July, 2001").
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.