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Independence polynomials of graphs

Takayuki Hibi, Selvi Kara, Dalena Vien

Abstract

In this paper, we study the independence polynomial $P_G(x)$ of a finite simple graph $G$, with emphasis on the evaluation at $x=-1$, symmetry, and its connection with the $h$-polynomial of the edge ideal of $G$. For big star graphs, we determine exactly when $P_G(-1)$ is $0, 1$, or $-1$, characterize the pseudo-Gorenstein$^*$ members, and show that there is a unique big star with symmetric independence polynomial. We also study graphs obtained from a graph $H$ by attaching leaves to selected vertices. We derive an explicit formula for the resulting independence polynomial, determine the corresponding value at $-1$, and prove that if every vertex of $H$ receives at least one leaf, then the independence polynomial is symmetric if and only if each vertex receives exactly two leaves. As an application, we obtain exact criteria for the values of $P_G(-1)$ and for the pseudo-Gorenstein$^*$ members of caterpillar graphs. For cochordal graphs, we classify all symmetric independence polynomials. Finally, for connected graphs on $n$ vertices with small independence numbers, we determine the exact range of possible values of $P_G(-1)$.

Independence polynomials of graphs

Abstract

In this paper, we study the independence polynomial of a finite simple graph , with emphasis on the evaluation at , symmetry, and its connection with the -polynomial of the edge ideal of . For big star graphs, we determine exactly when is , or , characterize the pseudo-Gorenstein members, and show that there is a unique big star with symmetric independence polynomial. We also study graphs obtained from a graph by attaching leaves to selected vertices. We derive an explicit formula for the resulting independence polynomial, determine the corresponding value at , and prove that if every vertex of receives at least one leaf, then the independence polynomial is symmetric if and only if each vertex receives exactly two leaves. As an application, we obtain exact criteria for the values of and for the pseudo-Gorenstein members of caterpillar graphs. For cochordal graphs, we classify all symmetric independence polynomials. Finally, for connected graphs on vertices with small independence numbers, we determine the exact range of possible values of .
Paper Structure (6 sections, 30 theorems, 117 equations)

This paper contains 6 sections, 30 theorems, 117 equations.

Table of Contents

  1. Preliminaries
  2. Big Stars
  3. Whiskering and Independence Polynomials
  4. An application: caterpillar graphs
  5. Independence polynomial of cochordal graphs
  6. Values of the independence polynomial at $x=-1$ for small $\alpha(G)$

Key Result

Proposition 1

Let $\alpha=\alpha(G)$. Then In particular, $h_{\alpha}(G)=(-1)^{\alpha}P_G(-1)$.

Theorems & Definitions (59)

  • Proposition 1
  • Theorem 1.1
  • Definition 1
  • Lemma 1
  • Theorem 1.2
  • Lemma 2
  • Definition 2
  • Lemma 3
  • proof
  • Remark 1
  • ...and 49 more