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Generalized degree polynomials of trees

Ricky Ini Liu, Michael Tang

Abstract

The generalized degree polynomial $\mathbf{G}_T(x,y,z)$ of a tree $T$ is an invariant introduced by Crew that enumerates subsets of vertices by size and number of internal and boundary edges. Aliste-Prieto et al. proved that $\mathbf{G}_T$ is determined linearly by the chromatic symmetric function $\mathbf{X}_T$, introduced by Stanley. We present several classes of information about $T$ that can be recovered from $\mathbf{G}_T$ and hence also from $\mathbf{X}_T$. Examples of such information include the double-degree sequence of $T$, which enumerates edges of $T$ by the pair of degrees of their endpoints, and the leaf adjacency sequence of $T$, which enumerates vertices of $T$ by degree and number of adjacent leaves. We also discuss a further generalization of $\mathbf{G}_T$ that enumerates tuples of vertex sets and show that this is also determined by $\mathbf{X}_T$.

Generalized degree polynomials of trees

Abstract

The generalized degree polynomial of a tree is an invariant introduced by Crew that enumerates subsets of vertices by size and number of internal and boundary edges. Aliste-Prieto et al. proved that is determined linearly by the chromatic symmetric function , introduced by Stanley. We present several classes of information about that can be recovered from and hence also from . Examples of such information include the double-degree sequence of , which enumerates edges of by the pair of degrees of their endpoints, and the leaf adjacency sequence of , which enumerates vertices of by degree and number of adjacent leaves. We also discuss a further generalization of that enumerates tuples of vertex sets and show that this is also determined by .

Paper Structure

This paper contains 16 sections, 13 theorems, 75 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The chromatic symmetric function and generalized degree polynomial
  3. Notation and conventions
  4. The chromatic symmetric function
  5. The generalized degree polynomial
  6. Trees with equal generalized degree polynomials
  7. Degree embeddings
  8. Degree embedding numbers from the generalized degree polynomial
  9. The rational functions $\mathbf{\Gamma}_T^{(k)}$
  10. Double-degree numbers
  11. Matchings and paths
  12. Triple-degree numbers
  13. Leaf adjacency
  14. Higher-order generalized degree polynomials
  15. Future work
  16. ...and 1 more sections

Key Result

Proposition 2.1

If $F$ is a forest of order $n$, then

Figures (3)

  • Figure 1: The smallest pair of trees with the same generalized degree polynomial.
  • Figure 2: An example illustrating the proof of Proposition \ref{['prop:gdp-augmentation']}. Solid black edges and black vertices are in $T$; dashed blue edges and smaller blue vertices are added to get $U$; circled vertices are in $A$. The parameters have values $(n,k) = (6,5)$, $(a,b,c) = (8,11,4)$, $(\ell, i) = (4,3)$, $(d,e) = (2,1)$, and $\ell^* = 20$.
  • Figure 3: The tree $T$ of Example \ref{['ex:degree-embeddings']}. Vertices are labeled with their degrees.

Theorems & Definitions (33)

  • Proposition 2.1: stanley
  • Definition 2.2
  • Example 2.3
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • proof : Proof of Proposition \ref{['prop:GT-linear']}
  • Proposition 2.6
  • proof
  • Definition 3.1
  • ...and 23 more