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

Research paper by **Hein van der Holst**

Indexed on: **25 Oct '12**Published on: **25 Oct '12**Published in: **Mathematics - Combinatorics**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### 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.