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

Abstract: Let $\mathbb{F}$ be an infinite field with characteristic different from two. For a graph $G=(V,E)$ with $V={1,...,n}$, let $S(G;\mathbb{F})$ be the set of all symmetric $n\times n$ matrices $A=[a_{i,j}]$ over $\mathbb{F}$ with $a_{i,j}\not=0$, $i\not=j$ if and only if $ij\in E$. We show that if $G$ is the complement of a partial $k$-tree and $m\geq k+2$, then for all nonsingular symmetric $m\times m$ matrices $K$ over $\mathbb{F}$, there exists an $m\times n$ matrix $U$ such that $U^T K U\in S(G;\mathbb{F})$. As a corollary we obtain that, if $k+2\leq m\leq n$ 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 $S(G;\reals)$ with $p$ positive and $q$ negative eigenvalues.