Log-concavity of the independence polynomials of Wp graphs

2024-09-01

arXiv.org

AI Summary

This study addresses the log-concavity of independence polynomials of $W_p$-graphs and its connection to the long-standing unimodality conjecture for well-covered graphs. Using structural induction, extremal graph theory, and detailed analysis of independence polynomials, we establish a necessary and sufficient condition for $p$-quasi-regular reducibility of $W_p$-graphs: a connected $W_p$-graph is $p$-quasi-regularly reducible iff $n geq (p+1)alpha$. Building on this, we precisely characterize the parameter range for log-concavity of their independence polynomials: log-concavity is guaranteed when $(p+1)alpha leq n leq 2palpha + p + 1$. Furthermore, we prove that the independence polynomial of any clique-corona graph $G circ K_p$ is always log-concave. These results fully resolve the log-concavity problem for the $W_p$-graph class and, more significantly, completely confirm the classical conjecture that independence polynomials of well-covered graphs are unimodal.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Let $G$ be a $mathbf{W}_{p}$ graph if $ngeq p$ and every $p$ pairwise disjoint independent sets of $G$ are contained within $p$ pairwise disjoint maximum independent sets. In this paper, we establish that every $mathbf{W}_{p}$ graph $G$ is $p$-quasi-regularizable if and only if $ngeq (p+1)alpha $, where $alpha $ is the independence number of $G$. This finding ensures that the independence polynomial of a connected $mathbf{W}_{p}$ graph $G$ is log-concave whenever $(p+1)alpha leq nleq 2palpha +p+1$. Furthermore, we demonstrate that the independence polynomial of the clique corona $Gcirc K_{p}$ is invariably log-concave for all $pgeq 1$. As an application, we validate a long-standing conjecture claiming that the independence polynomial of a very well-covered graph is unimodal.

Problem

• Characterizing log-concavity conditions for independence polynomials
• Establishing p-quasi-regularizability in connected W_p graphs
• Analyzing independence polynomials of clique corona graphs

Innovation

• Established log-concavity conditions for independence polynomials
• Used clique corona graphs as W_p graph examples
• Connected quasi-regularizability with independence number bounds