2016/05/07 by Saeid Alikhani, Alikhani, Saeid, Nasrin Jafari +1
Computer Science · Mathematics · #05C69 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.1605.02222
openalex publication_date 2016/05/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Let G = (V, E) be a simple graph of order n. The total dominating set of G is a subset D of V that every vertex of V is adjacent to some vertices of D. The total domination number of G is equal to minimum cardinality of total dominating set in G and denoted by γt(G). The total domination polynomial of G is the polynomial Dt(G,x)=∑i=γt(G)n dt(G,i), where dt(G,i) is the number of total dominating sets of G of size i. In this paper, we study roots of total domination polynomial of some graphs. We show that all roots of Dt(G, x) lie in the circle with center (-1, 0) and the radius √[δ]2n-1, where δ is the minimum degree of G. As a consequence we prove that if δ≥ (2n)/(3), then every integer root of Dt(G, x) lies in the set \-3,-2,-1,0\.