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Counting lattice points that appear as algebraic invariants of Cameron-Walker graphs

Sara Faridi, Iresha Madduwe Hewalage

Abstract

In 2021, Hibi et. al. studied lattice points in $\mathbb{N}^2$ that appear as $(\depth R/I,\dim R/I)$ when $I$ is the edge ideal of a graph on $n$ vertices, and showed these points lie between two convex polytopes. When restricting to the class of Cameron--Walker graphs, they showed that these pairs do not form a convex lattice polytope. In this paper, for the edge ideal $I$ of a Cameron--Walker graph on $n$ vertices, we find how many points in $\mathbb{N}^2$ appear as $(\depth(R/I),\dim(R/I))$, and how many points in $\mathbb{N}^4$ appear as $(\depth(R/I),\reg(R/I),\dim(R/I),\degh(R/I)).$

Counting lattice points that appear as algebraic invariants of Cameron-Walker graphs

Abstract

In 2021, Hibi et. al. studied lattice points in that appear as when is the edge ideal of a graph on vertices, and showed these points lie between two convex polytopes. When restricting to the class of Cameron--Walker graphs, they showed that these pairs do not form a convex lattice polytope. In this paper, for the edge ideal of a Cameron--Walker graph on vertices, we find how many points in appear as , and how many points in appear as
Paper Structure (5 sections, 18 theorems, 59 equations, 2 figures)

This paper contains 5 sections, 18 theorems, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. On the set ${\mathop{\mathrm{\mathop{\mathrm{Graph}}\nolimits_{\mathop{\mathrm{depth}}\nolimits,\dim}}}\nolimits(n)}$
  4. the Size of $\mathop{\mathrm{\mathop{\mathrm{CW}}\nolimits_{\mathop{\mathrm{depth}}\nolimits,\mathop{\mathrm{reg}}\nolimits,\dim,\mathop{\mathrm{deg h}}\nolimits}}}\nolimits(n)$
  5. Acknowledgment

Key Result

Theorem 2.2

k Let $G$ be a connected graph. Then $\mathop{\mathrm{m}}\nolimits(G)=\mathop{\mathrm{im}}\nolimits(G)$ if and only if $G$ is a star or a star triangle, or consists of a connected bipartite graph $B$ with vertex partition $\{u_1,\dots,u_m\} \cup \{v_1,\dots,v_n\}$ such that there is at least one lea

Figures (2)

  • Figure 1: A simple graph
  • Figure 3: A Cameron--Walker Graph

Theorems & Definitions (34)

  • Example 2.1
  • Theorem 2.2
  • Example 2.3
  • Example 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5: The size of $A$
  • proof
  • ...and 24 more