On the tree cover number and the positive semidefinite maximum nullity of a graph

TL;DR

This paper investigates the relationship between the tree cover number and the maximum positive semidefinite nullity of graphs, proving conjectures and establishing bounds for outerplanar graphs.

Contribution

It proves the conjecture that the tree cover number is less than or equal to the maximum positive semidefinite nullity for certain graph families and provides bounds for outerplanar graphs.

Findings

For connected outerplanar graphs, T(G) = M_+(G) ≤ ⌈n/2⌉.

For outerplanar graphs with no 3- or 4-cycle, T(G) = M_+(G) ≤ n/3.

Characterization of outerplanar graphs with T(G) = M_+(G) = ⌈n/2⌉.

Abstract

For a simple graph $G=(V,E),$ let $\mathcal{S}_+(G)$ denote the set of real positive semidefinite matrices $A=(a_{ij})$ such that $a_{ij}\neq 0$ if $\{i,j\}\in E$ and $a_{ij}=0$ if $\{i,j\}\notin E$. The maximum positive semidefinite nullity of $G$, denoted $\operatorname{M}_+(G),$ is $\max\{\operatorname{null}(A)|A\in \mathcal{S}_+(G)\}.$ A tree cover of $G$ is a collection of vertex-disjoint simple trees occurring as induced subgraphs of $G$ that cover all the vertices of $G$. The tree cover number of $G$, denoted $T(G)$, is the cardinality of a minimum tree cover. It is known that the tree cover number of a graph and the maximum positive semidefinite nullity of a graph are equal for outerplanar graphs, and it was conjectured in 2011 that $T(G)\leq M_+(G)$ for all graphs [Barioli et al., Minimum semidefinite rank of outerplanar graphs and the tree cover number, $ Elec. J. Lin. Alg.,$ 2011]. We show that the conjecture is true for certain graph families. Furthermore, we prove bounds on $T(G)$ to show that if $G$ is a connected outerplanar graph on $n\geq 2$ vertices, then $\operatorname{M}_+(G)=T(G)\leq \left\lceil\frac{n}{2}\right\rceil$, and if $G$ is a connected outerplanar graph on $n\geq 6$ vertices with no three or four cycle, then $\operatorname{M}_+(G)=T(G)\leq \frac{n}{3}$. We also characterize connected outerplanar graphs with $\operatorname{M}_+(G)=T(G)=\left\lceil\frac{n}{2}\right\rceil.$