Combinatorial and analytic aspects of independence polynomials of zero divisor graphs
For mathematicians studying graph polynomials and commutative rings, this provides a new class of graphs where the unimodal conjecture holds, but the result is incremental as it applies only to specific zero divisor graphs.
The paper proves that the independence polynomials of certain zero divisor graphs are unimodal and log-concave, confirming the unimodal conjecture for these graphs, and characterizes the zeros of these polynomials.
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.