Tuesday, October 8, 2013

Beta characteristic and isomorphism

Suppose you have any kind of graph:
Note that some of the edges are weighted, vertices can be connected to themselves, and it doesn't have to be planar. It could also be a directed graph, where the weight from one vertex to another is considered -k times the weight in the other direction for some constant k. A restriction is that if you are treating graphs with multiple edges between vertexes, the weight needs to reflect the number of edges, for example 3 edges between vertexes could be weight 3, and if it's also a directed graph with multiple edges the edges need to be in the same direction. Maybe it's also worth noting that by these conventions a weighted directed graph where all the weights are multiplied by -1 and all the directions are reversed are considered isomorphic.

Consider the labeling of the vertices for the graph pictured above S1:
This is a labeling that is the solution to the following equations:
Every equation is 1 + the sum of the weights of the connections to it's neighbors.

Note that there will always be a solution to this kind of set of equations because these are basically equations for n hyperplanes in n dimensions and it can be seen that none will be parallel because one would have to be a constant plus another but the 1+ on the right hand side prevents that. So the n hyperplanes will always meet at exactly one point, unless it ends up that two or more of the equations were exactly the same, in which case the variables that are free in the set of equations can be set to 0 to get one point.
It is the same equations as before except the non 1+ part on the right hand side is divided by 2. For this graph the solution S2 is:

Basically the 1+ on the right hand side of the equations makes sure that equations 1 and 2 aren't linearly related.
Finally make a matrix whose rows and columns are formed by these solutions S1..SN for a graph of n vertices and sort this matrix by exchanging columns until the first row is in increasing order.  This gives you what I call the Beta characteristic B(i, j). Note that however the vertices were named in the original graph yields the same Beta characteristic after sorting.
So for a graph of n vertices, I will claim that two graphs are isomorphic if and only if this Beta characteristic is the same.

First that a Beta characteristic exists and corresponds to one graph:
Let B be the beta characteristic and m(i,j) be variables.
For a graph of n vertices, there are n^2 equations (i,j) and n^2 variables m(i,j)
B(i, j) = 1+(m(1,j)B(i,1)+m(2, j)B(i,2)+...m(n,j)*B(i,n))/i
All of these equations are linearly independent, because every equation has a constant term of 1, no amount of them added together can equal another. And because only the variable part of each equation is divided by i, you can't multiply any two equations by constants and make the same equation. For any choice of m(i,j) there is a unique Beta characteristic B because the equations with left hand side B(i, x) are n hyperplanes in n dimensional space which meet at only one point. Each such point becomes a row in a matrix and these are sorted to become B. For a choice of B there is a unique solution to the m(i,j)'s because there are the same number of equations as variables and they are all linearly independent. These m(i,j) are translated in a unique way to a corresponding graph. .
Second that isomorphic graphs correspond to one Beta characteristic. Isomorphic graphs are only different in the way the vertices are labeled and any labeling of the same graph results in the same Beta characteristic as the columns of the Beta characteristic are always sorted in increasing order of the first row from left to right.

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