Speaker
Description
Given a simple graph $G$, $\mathcal{S}(G)$ is the set of real symmetric matrices indexed by the vertices in $G$ and with off-diagonal zeros corresponding to non-edges in $G$. The problem of finding the maximum nullity of a matrix in $\mathcal{S}(G)$ has been extensively studied. We consider the maximum nullity of a matrix $A$ and its principal submatrix $A(i)$ corresponding to deleting the vertex $i$. The strong Arnold property possessed by some matrices is one of the useful tools for studying maximum nullity. The parameter $\xi(G)$ is the maximum nullity of a matrix in $\mathcal{S}(G)$ with the strong Arnold property, and has been shown to be minor monotone. We defined a new parameter which we call $\xi\xi(G)$ for the maximum sum of the nullities of $A$ and $A(i)$ for matrices $A \in \mathcal{S}(G)$. We show that $\xi\xi(G)$ is also minor monotone. In this talk I will show how $\xi\xi(G)$ is a refinement of $\xi(G)$ and minimal minors for small values of $\xi\xi(G)$.