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A strong nullity parameter for rooted graphs

Aida Abiad, Mary Flagg, H. Tracy Hall, Jephian C. -H. Lin, Bryan Shader

Abstract

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 $ξξ(G,i)$ 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 $ξξ(G,i) \geq s$ 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 $ξ(G) \geq k$.

A strong nullity parameter for rooted graphs

Abstract

The inverse eigenvalue problem of a graph studies the possible spectra of matrices associated with , 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 on vertices with a designated root vertex, we investigate all possible nullity pairs where the first nullity is that of an symmetric matrix associated to and the second nullity is that of the principal submatrix of size that results from deleting the row and column associated to the root vertex. We define a new parameter for rooted graphs 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 for by finding the minimal minors for each of these families. These families turn out to have strong connections to the minimal minors for .
Paper Structure (8 sections, 28 theorems, 50 equations, 4 figures)

This paper contains 8 sections, 28 theorems, 50 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bifurcation Lemma
  4. Extending the root
  5. Schur complement
  6. Minimal minors for xixi(G, i) >= 4,5
  7. Many aspects of i-SNIP
  8. Concluding remarks

Key Result

Proposition 2.1

Let $(G,i)$ be a rooted graph. Then $(G,i)$ allows the nullity pair $(k,k)$ (respectively, with $i$-SNIP) if and only if $(G,i)$ allows the nullity pair $(k+1, k)$ (respectively, with $i$-SNIP).

Figures (4)

  • Figure 1: Minimal minors for $\xi\xi$.
  • Figure 2: The nullity pair staircase.
  • Figure 3: The $T_3$-family with all possible roots marked as filled vertices.
  • Figure 4: An illustration of the proof of \ref{['lem:notcut']}.

Theorems & Definitions (51)

  • Proposition 2.1: Corollaries 2.3 and 3.10 of SNIP
  • Theorem 2.2: Theorem 3.9 of SNIP
  • Proposition 2.3
  • Theorem 3.1
  • proof
  • Theorem 3.2: Bifurcation Lemma
  • Lemma 3.3: West Lemma
  • proof
  • Lemma 3.4: South Lemma
  • proof
  • ...and 41 more