Daily Archives: October 28, 2021

Solution: 2021-18 Independent sets in a tree

Let \(T\) be a tree (an acyclic connected graph) on the vertex set \([n]=\{1,\dots, n\}\).
Let \(A\) be the adjacency matrix of \(T\), i.e., the \(n\times n\) matrix with \(A_{ij} = 1\) if \(i\) and \(j\) are adjacent in \(T\) and \(A_{ij}=0\) otherwise. Prove that the number of nonnegative eigenvalues of \(A\) equals to the size of the largest independent set of \(T\). Here, an independent set is a set of vertices where no two vertices in the set are adjacent.

The best solution was submitted by 전해구 (기계공학과 졸업생, +4). Congratulations!

Here is the best solution of problem 2021-18.

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