[Paper Review] Very well-covered graphs with log-concave independence polynomials

Vadim E. Levit, Eugen Mǎndrescu|arXiv (Cornell University)|Nov 10, 2004

Advanced Combinatorial Mathematics9 references20 citations

TL;DR

This paper demonstrates that the independence polynomial of very well-covered graphs derived from trees with stability number ≤3 or specific path/spider graphs (e.g., $K_{1,n}$, $P_n$) is log-concave, implying unimodality. The key contribution is proving that $I(G^*;x)$ is log-concave for such graphs, supporting the long-standing conjecture that independence polynomials of well-covered and forest graphs are unimodal.

ABSTRACT

If for any $k$ the $k$-th coefficient of a polynomial $I(G;x)$ is equal to the number of stable sets of cardinality $k$ in the graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). Alavi, Malde, Schwenk and Erdos (1987) conjectured that $I(G;x)$ is unimodal, whenever $G$ is a forest, while Brown, Dilcher and Nowakowski (2000) conjectured that $I(G;x)$ is unimodal for any well-covered graph G. Michael and Traves (2003) showed that the assertion is false for well-covered graphs with $a(G)$ > 3 ($a(G)$ is the size of a maximum stable set of the graph $G$), while for very well-covered graphs the conjecture is still open. In this paper we give support to both conjectures by demonstrating that if $a(G)$ < 4, or $G$ belongs to ${K_{1,n}, P_{n}: n > 0}$, then $I(G*;x)$ is log-concave, and, hence, unimodal (where $G*$ is the very well-covered graph obtained from $G$ by appending a single pendant edge to each vertex).

Motivation & Objective

• To support the conjecture that independence polynomials of well-covered graphs are unimodal, particularly for very well-covered graphs.
• To investigate the log-concavity of independence polynomials in very well-covered graphs constructed from trees with small stability number.
• To extend previous results on unimodality to a broader class of very well-covered graphs using structural graph properties.
• To explore the relationship between graph structure (e.g., trees, spiders, paths) and the log-concavity of their independence polynomials.

Proposed method

• Constructs very well-covered graphs $G^*$ by appending a pendant edge to each vertex of a base graph $G$.
• Applies recursive decomposition via Proposition 1: $I(G;x) = I(G-v;x) + xI(G-N[v];x)$ to derive independence polynomials.
• Uses the fact that the product of log-concave polynomials is log-concave (Theorem 1) and that log-concavity implies unimodality.
• Establishes closed-form expressions for independence polynomials of specific families: $W_n$ (centipedes), $\bigtriangleup_n$, and $\bigtriangleup_n \circleddash K_2$.
• Employs induction on the number of triangles in the graph to prove that $I(W_n;x)$ is log-concave for all $n$.
• Leverages known results on real-rootedness of independence polynomials (e.g., for paths and claw-free graphs) to support structural insights.

Experimental results

Research questions

• RQ1 Is the independence polynomial of any very well-covered graph derived from a tree with $\alpha(G) \leq 3$ log-concave?
• RQ2 Does the independence polynomial of $G^*$ remain log-concave when $G$ is a path $P_n$ or a star $K_{1,n}$?
• RQ3 Can the log-concavity of independence polynomials be established for infinite families of very well-covered graphs?
• RQ4 To what extent does log-concavity of $I(G;x)$ imply unimodality in very well-covered graphs?
• RQ5 Are there structural graph classes (e.g., spiders, claw-free graphs) for which the independence polynomial is guaranteed to be log-concave?

Key findings

• The independence polynomial $I(G^*;x)$ is log-concave for any graph $G$ with $\alpha(G) \leq 3$, including $G = K_{1,n}$ and $G = P_n$.
• For centipedes $W_n$, the independence polynomial satisfies $I(W_n;x) = (1+x)^{n} \cdot I(\bigtriangleup_n;x)$, and is proven to be log-concave.
• The independence polynomial of $W_n$ is log-concave for all $n$, as shown via induction and recursive decomposition using Proposition 1.
• The product of log-concave polynomials remains log-concave, and since $I(G;x)$ is log-concave for base graphs with $\alpha(G) \leq 3$, $I(G^*;x)$ inherits this property.
• The independence polynomial of $\bigtriangleup_n$ and $\bigtriangleup_n \circleddash K_2$ is log-concave, which underpins the proof for $W_n$.
• The result supports the broader conjecture that independence polynomials of all well-covered forests are unimodal, as log-concavity implies unimodality.