A strong nullity parameter for rooted graphs

The inverse eigenvalue problem of a graph $G$ studies the possible spectra of matrices associated with $G$, including as an important subproblem the possible nullities of such a matrix. Much research in this area to date has focused only on the spectrum of the matrix itself, but there are applications of inverse eigenvalue problems that also involve the interaction between that spectrum and the spectrum of some maximal proper principal submatrix, or in other words the interlacing spectrum that results from crossing out any one row and the same column. Motivated by this refined information, given a graph $G$ on $n$ vertices with a designated root vertex, we investigate all possible nullity pairs where the first nullity is that of an $n \times n$ symmetric matrix associated to $G$ and the second nullity is that of the principal submatrix of size $(n - 1) \times (n - 1)$ that results from deleting the row and column associated to the root vertex. We define a new parameter $\xi\xi(G,i)$ for rooted graphs $(G,i)$ equipped with the strong nullity interlacing property that coordinates the two values of a nullity pair, and we show that this graph parameter is minor monotone. Moreover, we prove a bifurcation lemma for the strong nullity interlacing property. We use these new tools to characterize the rooted graphs with $\xi\xi(G,i) \geq s$ for $s \in \{ 0,1,2,3,4, 5\}$ by finding the minimal minors for each of these families. These families turn out to have strong connections to the minimal minors for $\xi(G) \geq k$.