The Kirchhoff matrix of a graph can be obtained by getting the difference between the degree matrix and the adjacency matrix of the graph. What properties of the graph can be described by investigating the Kirchhoff matrix?
One interesting property is that if you divide each absolute value of that matrix by the diagonal value in the same row (correspondent to the node degree) you get the weight of each node link or edge over all links or edges of that node (assuming there are no self-incident nodes). Such metric could be a measure of the importance of the edge to the node connectivity.
Kind regards.
Fernando.
Thank you very much Dr. Fernando. That's an awesome answer.
You can detrmine the number of spanning trees of the given graph.
See Kirchoff Theorem.
Best regards and good luck!
The smallest eigenvalue of L is 0. The second smallest eigenvalue is known as the "Algebraic connectivity" of the graph, and gives a measure of how connected the graph is. The corresponding eigenvector is called the "Fiedler vector" and is widely used in heuristics for the "Edge separator problem", which is to partition the graph into sets so that the number of edges between sets is minimized.
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