The inverse inertia problem for the complements of partial k -trees

Abstract

Let F be an infinite field with characteristic different from two. For a graph G = (V, E) with V = {1, . . . , n}, let S(G; F) be the set of all symmetric n × n matrices A = [a i,j ] over F with a i,j = 0, i = j if and only if ij ∈ E. We show that if G is the complement of a partial k-tree and m ≥ k + 2, then for all nonsingular symmetric m × m matrices K over F, there exists an m × n matrix U such that U T KU ∈ S(G; F). As a corollary we obtain that, if k + 2 ≤ m ≤ n and G is the complement of a partial k-tree, then for any two nonnegative integers p and q with p + q = m, there exists a matrix in S(G; R) with p positive and q negative eigenvalues.