On the roots of total domination polynomial of graphs. (18th May 2020)

• Record Type: Journal Article
• Authors: Jafari, Nasrin; Alikhani, Saeid
• Is Part Of: Journal of discrete mathematical sciences & cryptography. Volume 23:Number 4(2020)
• Page Start: 795
• Page End: 807
• Publication Date: 2020-05-18
• Subjects: 05C69; Total domination polynomial -- Total dominating set -- Root; Computer science -- Mathematics -- Periodicals; Cryptography -- Periodicals; Computer science -- Mathematics; Cryptography; Periodicals; 004.0151
• DOI: 10.1080/09720529.2019.1616908
• Languages: English
• ISSNs: 0972-0529

Abstract

Abstract: Let G = ( V, E ) be a simple graph of order n . A total dominating set of G is a subset D of V, such that every vertex of V is adjacent to at least one vertex in D . The total domination number of G is minimum cardinality of total dominating set in G and is denoted by γt ( G ). The total domination polynomial of G is the polynomial, where dt ( G, i ) is the number of total dominating sets of G of size i . In this paper, we study roots of the total domination polynomial of some graphs. We show that all roots of Dt ( G, x ) lie in the circle with center (–1, 0) and radius, where d is the minimum degree of G . As a consequence, we prove that if, then every integer root of D t ( G, x ) lies in the set {–3, –2, –1, 0}.