Ela the Combinatorial Inverse Eigenvalue Problem: Complete Graphs and Small Graphs with Strict Inequality∗

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 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 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.