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Some new results on domination roots of a graph

Authors
Journal
Electronic Notes in Discrete Mathematics
1571-0653
Publisher
Elsevier
Publication Date
Volume
43
Identifiers
DOI: 10.1016/j.endm.2013.07.062
Keywords
  • Domination Polynomial
  • Domination Root
  • Triangle
  • Complex Root
  • Lexicographic Product
Disciplines
  • Linguistics

Abstract

Abstract Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,λ)=∑i=0nd(G,i)λi, where d(G, i) is the number of dominating sets of G of size i. Every root of D(G,λ) is called the domination root of G. We present families of graphs whose their domination polynomial have no nonzero real roots. We observe that these graphs have complex domination roots with positive real part. Then, we consider the lexicographic product of two graphs and obtain a formula for domination polynomial of this product. Finally, we construct a family of graphs which their domination roots are dense in all of C.

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