O ct 2 01 2 The inverse inertia problem for the complements of partial k - trees

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 = [ai,j ] over F with ai,j 6= 0, i 6= 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 UKU ∈ 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. keywords: graph, inertia, symmetric, graph complement, treewidth MSC: 05C05, 15A03