Efficient Computation of the Characteristic Polynomial of a Threshold Graph

Abstract

Abstract. An efficient algorithm is presented to compute the characteristic polynomial of a threshold graph. Threshold graphs were introduced by Chvátal 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.

The role of the anti-regular graph in the spectral analysis of threshold graphs

“…Moreover, in [11] the authors determine the threshold graph with smallest negative eigenvalue and show that all eigenvalues of a threshold graph are simple except possibly −1 and/or 0. In [12], the authors present an O(n 2 ) algorithm for computing the characteristic polynomial of an n-vertex threshold graph and an improved algorithm running in almost linear time was constructed in [7]. In [13], it is proved that no threshold graph has an eigenvalue in the interval (−1, 0) and a study of noncospectral equienergetic threshold graphs was undertaken.…”

An explicit formula for the distance characteristic polynomial of threshold graphs

“…There is a considerable body of knowledge on the spectral properties of threshold graphs related to adjacency matrix [3,8,9,14,15,16,20,22,23].…”

No threshold graphs are cospectral

“…increasing sequences alternating even and odds numbers such that the last term has the same than parity n. For instanceI 7,4 = {(2, 3, 4, 5),(2,3,4,7),(2,3,6,7),(2,5,6,7), (4, 5, 6, 7)},…”