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THE COMBINATORIAL INVERSE EIGENVALUE PROBLEM: COMPLETE GRAPHS AND SMALL GRAPHS WITH STRICT INEQUALITY
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| Electronic Journal of Linear Algebra |
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Let G be a simple undirected graph on n vertices and let S(G) be the class of real symmetric n × n matrices whose nonzero off-diagonal entries correspond exactly to the edges of G. Given 2n − 1 real numbers 1 μ1 2 μ2 · · · n−1 μn−1 n, and a vertex v of G, the question is addressed of whether or not there exists A 2 S(G) with eigenvalues 1, . . . , n such that A(v) has eigenvalues μ1, . . . , μn−1, where A(v) denotes the matrix with the vth row and column deleted. General results that apply to all connected graphs G are given first, followed by a complete answer to the question for Kn. Since the answer is constructive it can be implemented as an algorithm; a Mathematica code is provided to do so. Finally, for all connected graphs on 4 vertices it is shown that the answer is affirmative if all six inequalities are strict.
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Graph, Interlacing inequalities, Inverse eigenvalue problem, Symmetric matrix.
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