[Paper Review] A Family of Well-Covered Graphs with Unimodal Independence Polynomials

Vadim E. Levit, Eugen Mǎndrescu|ArXiv.org|Jul 1, 2003

Graph theory and applicationsMathematics8 references16 citations

TL;DR

This paper proves that the independence polynomial of $G^*$, the graph formed by attaching a pendant edge to each vertex of $G$, is unimodal whenever $\alpha(G) \leq 4$. The authors establish a coefficient transformation formula linking $I(G;x)$ and $I(G^*;x)$, and use combinatorial inequalities to show unimodality for all such $G$, supporting broader conjectures on unimodal independence polynomials in well-covered graphs.

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 graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). J. I. Brown, K. Dilcher and R. J. Nowakowski (2000) conjectured that the independence polynomial of a well-covered graph $G$ (i.e., a graph whose all maximal independent sets are of the same size) is unimodal, that is, there exists an index $k$ such that the part of the sequence of coefficients from the first to $k$-th is non-decreasing while the other part of coefficients is non-increasing. T. S. Michael and N. Traves (2002) provided examples of well-covered graphs whose independence polynomials are not unimodal. A. Finbow, B. Hartnell and R. J. Nowakowski (1993) proved that under certain conditions, any well-covered graph equals G* for some $G$, where G* is the graph obtained from $G$ by appending a single pendant edge to each vertex of $G$. Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erdös (1987) asked whether for trees the independence polynomial is unimodal. V. E. Levit and E. Mandrescu (2002) validated the unimodality of the independence polynomials of some well-covered trees (e.g., $P_{n}^{*},K_{1,n}^{*}$, where $P_{n}$ is the path on $n$ vertices and $K_{1,n}$ is the $n$-star graph). In this paper we show that for any graph $G$ with the stability number alpha(G) < 5, the independence polynomial of G* is unimodal.

Motivation & Objective

• To investigate the unimodality of independence polynomials in well-covered graphs, particularly those of the form $G^*$, where each vertex of $G$ is given a pendant edge.
• To resolve a long-standing conjecture about the unimodality of independence polynomials in trees and well-covered graphs by analyzing a specific family of graphs.
• To provide a constructive proof that $I(G^*;x)$ is unimodal for all graphs $G$ with independence number $\alpha(G) \leq 4$, using coefficient transformation and combinatorial inequalities.

Proposed method

• Derive a closed-form transformation between the coefficients of $I(G;x)$ and $I(G^*;x)$ using binomial coefficients and stable set counts.
• Express $t_k$, the coefficient of $x^k$ in $I(G^*;x)$, as a linear combination of $s_i$ (coefficients of $I(G;x)$) weighted by binomial coefficients: $t_k = \sum_{i=0}^{\min(k,4)} \binom{n - i}{k - i} s_i$.
• Establish unimodality by proving $t_0 \leq t_1 \leq \cdots \leq t_m$ and $t_{m+2} \geq t_{m+3} \geq \cdots \geq t_{2m}$ for $m = \lfloor \alpha(G^*)/2 \rfloor$, using inequalities on binomial coefficients.
• Analyze the middle coefficient difference $2t_{m+1} - t_m - t_{m+2}$ and show it is non-negative by expressing it as a sum with non-negative coefficients, ensuring unimodality.
• Handle small cases ($n \leq 4$) separately, verifying unimodality directly for $G = \sqcup 4K_1$ via $(1+2x)^4$.
• Use known results on very well-covered graphs and the fact that $G^*$ is always very well-covered to contextualize the findings within broader graph families.

Experimental results

Research questions

• RQ1 Is the independence polynomial of $G^*$ unimodal for all graphs $G$ with $\alpha(G) \leq 4$?
• RQ2 Can the unimodality of $I(G^*;x)$ be guaranteed under structural constraints on $G$, such as bounded independence number?
• RQ3 Does the transformation from $I(G;x)$ to $I(G^*;x)$ preserve unimodality when $\alpha(G) \leq 4$, despite $I(G;x)$ potentially being non-unimodal?
• RQ4 What are the possible locations of the mode of $I(G^*;x)$, and can they be characterized by $n = |V(G)|$?
• RQ5 Is the unimodality of $I(G^*;x)$ extendable to graphs with $\alpha(G) \geq 5$?

Key findings

• For all graphs $G$ with $\alpha(G) \leq 4$, the independence polynomial $I(G^*;x)$ is unimodal, as proven via coefficient inequalities and binomial coefficient analysis.
• The mode of $I(G^*;x)$ lies in $\{m, m+1, m+2\}$ where $m = \lfloor \alpha(G^*)/2 \rfloor$, and can be $\lfloor (n+1)/2 \rfloor$, $\lfloor (n+1)/2 \rfloor + 1$, or $\lfloor (n+1)/2 \rfloor + 2$ depending on $n$ and $G$.
• The polynomial $I((\sqcup 4K_1)^*;x) = (1+2x)^4 = 1 + 8x + 24x^2 + 32x^3 + 16x^4$ is unimodal with mode 3, confirming the result for the base case.
• The transformation $I(G^*;x)$ preserves unimodality for $\alpha(G) \leq 4$, even when $I(G;x)$ is not unimodal, as shown by counterexamples with $\alpha(G) = 4$ but non-unimodal $I(G;x)$.
• The authors construct connected non-tree very well-covered graphs $H = G^*$ with $\alpha(H) = n$ and unimodal $I(H;x)$ for any $n \geq 3$, proving Corollary 2.3.
• The paper leaves open the question of whether $I(G^*;x)$ remains unimodal for $\alpha(G) \geq 5$, suggesting a natural direction for future research.