My favorite application using eigenvalues: Eigenvalues and the Graham-Pollak Theorem
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چکیده
The famous Graham-Pollak Theorem states that one needs at least n− 1 complete bipartite subgraphs to partition the edge set of the complete graph on n vertices. Originally proved in conjunction with addressing for networking problems, this theorem is also related to perfect hashing and various questions about communication complexity. Since it’s original proof using Sylvester’s Law of Intertia, many other proofs have been discovered. Though the statement is purely combinatorial in nature, it is a surprising fact that most proofs have been algebraic. In this essay, we give a beautiful result about how the eigenvalues of an adjacency matrix of a graph relate to the minimum number of bipartite subgraphs necessary to partition its edge set.
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تاریخ انتشار 2013