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An $O(n^2)$ Algorithm for the Characteristic Polynomial of a Tree
We describe an algorithm, that uses O(n) arithmetic operations, for computing the determinant of the matrix M = (A + αI), where A is the adjacency matrix of an order n tree. Combining this algorithm with interpolation, we derive a simple algorithm requiring O(n 2) arithmetic operations, to find the characteristic polynomial of the adjacency matrix of any tree. We apply our algorithm and recompute a 22-degree characteristic polynomial, which had been incorrectly reported in the quantum chemistry literature. keywords: tree, adjacency matrix, characteristic polynomial.