Coulson-type integral formulas for the general energy of polynomials with real roots

Lu Qiao, Shenggui Zhang and Jing Li

Applied Mathematics and Computation, 2018, vol. 320, issue C, 202-212

Abstract: The energy of a graph is defined as the sum of the absolute values of its eigenvalues. In 1940 Coulson obtained an important integral formula which makes it possible to calculate the energy of a graph without knowing its spectrum. Recently several Coulson-type integral formulas have been obtained for various energies and some other invariants of graphs based on eigenvalues. For a complex polynomial ϕ(z)=∑k=0nakzn−k=a0∏k=1n(z−zk) of degree n and a real number α, the general energy of ϕ(z), denoted by Eα(ϕ), is defined as ∑zk≠0|zk|α when there exists k0∈{1,2,…,n} such that zk0≠0, and 0 when z1=⋯=zn=0. In this paper we give Coulson-type integral formulas for the general energy of polynomials whose roots are all real numbers in the case that α∈Q. As a consequence of this result, we obtain an integral formula for the 2l-th spectral moment of a graph. Furthermore, we show that our formulas hold when α is an irrational number with 0 < |α| < 2 and do not hold with |α| > 2.

Keywords: Graph energy; Coulson integral formula; General energy of polynomials (search for similar items in EconPapers)
Date: 2018
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0096300317306458
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:320:y:2018:i:c:p:202-212

DOI: 10.1016/j.amc.2017.09.024

Access Statistics for this article

Applied Mathematics and Computation is currently edited by Theodore Simos

More articles in Applied Mathematics and Computation from Elsevier
Bibliographic data for series maintained by Catherine Liu ().