Efficient Computation of the Characteristic Polynomial of a Threshold Graph

TL;DR

This paper introduces an efficient algorithm with $O(n \, \log^2 n)$ complexity for computing the characteristic polynomial of threshold graphs, significantly faster than previous quadratic-time methods.

Contribution

The paper presents a novel, faster algorithm for calculating the characteristic polynomial of threshold graphs, improving computational efficiency from quadratic to near-linearithmic time.

Findings

Achieved $O(n \log^2 n)$ running time for the algorithm.

Compared the new algorithm's efficiency against previous quadratic-time methods.

Validated the algorithm's correctness and performance on threshold graphs.

Abstract

An efficient algorithm is presented to compute the characteristic polynomial of a threshold graph. Threshold graphs were introduced by Chv\'atal and Hammer, as well as by Henderson and Zalcstein in 1977. A threshold graph is obtained from a one vertex graph by repeatedly adding either an isolated vertex or a dominating vertex, which is a vertex adjacent to all the other vertices. Threshold graphs are special kinds of cographs, which themselves are special kinds of graphs of clique-width 2. We obtain a running time of $O(n \log^2 n)$ for computing the characteristic polynomial, while the previously fastest algorithm ran in quadratic time. Keywords: Efficient Algorithms, Threshold Graphs, Characteristic Polynomial.