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...

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