## Thursday, August 15, 2013

### Examining the delta characteristic

A month ago or so I defined on my blog that the delta characteristic for a graph such as:
will be the solution to the set of equations of the form:
For example the first equation is A=(B+E+F)^(1/2) because A is connected to B,E, and F.
Notice that the solution's for A and F are the same and B and D are the same. And that the graph  is symmetric with respect to A and F and also B and D.
But since the labeling of the graph is arbitrary, I usually write the delta characteristic as an ordered list of solutions:
{2.452592664,3.007605387,3.007605387,3.098500103,3.098500103,3.494597399}

I've proven by exhaustion* that for simple graphs of 5 vertices this delta characteristic is different for any two graphs that are not isomorphic. And that it is the same for graphs that are isomorphic. But I'd like to know if this is true in general, if this is always a way to test for isomorphism of graphs.

* Proof by exhaustion for the case of 5 vertices:
G1:  1     1     1.259 1.259 1.587
G2:  1     1     2     2     2
G3:  1.341 1.341 1.799 1.799 1.897
G4:  1.375 1.484 1.484 1.892 2.205
G5:  1.587 1.587 1.587 1.587 2.519
G6:  1.587 1.587 2.245 2.52  2.52
G7:  1.673 1.673 2.246 2.246 2.799
G8:  1,406 1.979 2.161 2.161 2.510
G9:  1.415 2.005 2,259 2.505 2.601
G10: 2     2     2     2     2
G11: 1,640 2.368 2.692 2.804 2.804
G12: 1.751 2.425 2.425 2.814 3.068
G13: 2.213 2.213 2.318 2.686 2.686
G14: 2.289 2.289 2.289 2.620 2.620
G15: 2.314 2.314 2.314 2.314 3.042
G16: 1.823 3.079 3.079 3.079 3.326
G17: 2.579 2.579 2.579 3.326 3.326
G18: 2.400 2.881 2.881 2.952 2.952
G19: 2.503 2.503 2.961 2.961 3.306
G20: 2.667 3.213 3.213 3.556 3.556
G21: 3.130 3.130 3.130 3.130 3.538
G22: 3.368 3.368 3.781 3.781 3.781
G23: 4     4     4     4     4
(All possible simple graphs have a different characteristic... )

**Questions**
Now clearly all variables being equal to 0 is one solution to this type of system of equations.  But is it true that there is always one other solution that takes the form of positive values for all the variables? It is clear that there are always going to be the same number of equations as variables and that no one is a linear combination of any number of others or a constant multiple of another...