27th Conference of the International Linear Algebra Society (ILAS 2026)

Multi-digraphs with maximum nullity at most one

Not scheduled

25m

Minisymposium Talk The Inverse Eigenvalue Problem of a Graph and Zero Forcing The Inverse Eigenvalue Problem of a Graph and Zero Forcing

Speaker

Dr Hein Van der Holst (Georgia State University)

Description

For a multi-digraph $D$ with vertex-set $V=\{1,\ldots,n\}$ and arc-set $A$, let $Q(D)$ be the set of all real $n\times n$ matrices $A=[a_{i,j}]$ with $a_{i,j}\not=0$ if $i\not=j$ and there is a single arc from $i$ to $j$, $a_{i,j}\in \mathbb{R}$ if $i\not=j$ and there are multiple arcs from $i$ to $j$, $a_{i,j}=0$ if $i\not=j$ and there is no arc from $i$ to $j$, $a_{i,i}\not=0$ if there is no loop at vertex $i$, and $a_{i,i}\in \mathbb{R}$ if there is at least one loop at vertex $i$. The maximum nullity $M(D)$ of a multi-digraph $D$ is the maximum nullity of any $A\in Q(D)$. In this talk, we show progress on the characterization of multi-digraphs $D$ with $M(D)\leq 1$.

Joint work with Marina Arav and Cherine Fons