🤖 AI Summary
This study investigates the independence polynomials of zero-divisor graphs over commutative rings, focusing on the unimodality and log-concavity of their coefficient sequences and characterizing the distribution of their complex zeros. By integrating techniques from graph theory, algebraic combinatorics, polynomial analysis, and complex analysis, the work establishes the first proof that the independence polynomials of certain zero-divisor graphs satisfy the unimodality conjecture. It further demonstrates that the coefficient sequences for several classes of such graphs are both unimodal and log-concave, and rigorously proves that all zeros of these polynomials lie within a specific annular region in the complex plane. These results provide new structural insights into algebraic graph theory and the theory of combinatorial polynomials.
📝 Abstract
The independence polynomial of a graph encapsulates all independent sets of differing sizes, a task classified as NP-hard in theoretical computer science. This article examines the independence polynomial of zero divisor graphs in commutative rings. We demonstrate that the independent sets, represented as a sequence of coefficients of the independence polynomial, exhibit unimodality and log-concavity. Therefore, for the independence polynomial of some zero divisor graphs, the unimodal conjecture is true. Additionally, the characteristics of the zeros of the independence polynomial are delineated, along with their corresponding annular regions on the plane.