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Let G be an undirected graph on n vertices and let S(G) be the set of allreal symmetric n x n matrices whose nonzero off-diagonal entries occur inexactly the positions corresponding to the edges of G. The inverse inertiaproblem for G asks which inertias can be attained by a matrix in S(G). We givea complete answer to this question for trees in terms of a new family of graphparameters, the maximal disconnection numbers of a graph. We also give aformula for the inertia set of a graph with a cut vertex in terms of inertiasets of proper subgraphs. Finally, we give an example of a graph that is notinertia-balanced, and investigate restrictions on the inertia set of any graph.

The inverse inertia problem for graphs