The combinatorial inverse eigenvalue problem II: all cases for small graphs

Abstract. 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 to the edges of G. Given 2n − 1 real numbers λ1 ≥ μ1 ≥ λ2 ≥ μ2 ≥ · · · ≥ λn−1 ≥ μn−1 ≥ λn, and a vertex v of G, the question is addressed of whether or not there exists A ∈ S(G) with eigenvalues λ1, . . . , λn such that A(v) has eigenvalues μ1, . . . , μn−1, where A(v) denotes the matrix with vth row and column deleted. A complete solution can be given for the path on n vertices with v a pendant vertex and also for the star on n vertices with v the dominating vertex. The main result is a complete solution to this "λ, μ" problem for all connected graphs on 4 vertices.

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