1,040 research outputs found
Some families of graphs with no nonzero real domination roots
Let G be a simple graph of order n. The domination polynomial is the
generating polynomial for the number of dominating sets of G of each
cardinality. A root of this polynomial is called a domination root of G.
Obviously 0 is a domination root of every graph G. In the study of the
domination roots of graphs, this naturally raises the question: which graphs
have no nonzero real domination roots? In this paper we present some families
of graphs whose have this property.Comment: 13 pages, 5 figures. arXiv admin note: text overlap with
arXiv:1401.209
On the Domination Polynomials of Friendship Graphs
Let be a simple graph of order . The {\em domination polynomial} of
is the polynomial , where
is the number of dominating sets of of size .
Let be any positive integer and be the Friendship graph with vertices and edges, formed by the join of with . We
study the domination polynomials of this family of graphs, and in particular
examine the domination roots of the family, and find the limiting curve for the
roots. We also show that for every , is not
-unique, that is, there is another non-isomorphic graph with the
same domination polynomial. Also we construct some families of graphs whose
real domination roots are only and . Finally, we conclude by discussing
the domination polynomials of a related family of graphs, the -book graphs
, formed by joining copies of the cycle graph with a common
edge.Comment: 16 pages, 7 figures. New version of paper entitled "On
-equivalence class of friendship graphs
Domination polynomials of k-tree related graphs
Let be a simple graph of order . The domination polynomial of is
the polynomial , where is
the number of dominating sets of of size and is the
domination number of . In this paper we study the domination polynomials of
several classes of -tree related graphs. Also, we present families of these
kind of graphs, whose domination polynomial have no nonzero real roots
An atlas of domination polynomials of graphs of order at most six
The domination polynomial of a graph of order is the polynomial
, where is the number of
dominating sets of of size , and is the domination number of
. The roots of domination polynomial is called domination roots. In this
article, we compute the domination polynomial and domination roots of all
graphs of order less than or equal to 6, and show them in the tables.Comment: This atlas has published in the PhD thesis of the first author in
2009 and also in book "Dominating sets and domination polynomials of graphs:
Domination polynomial: A new graph polynomial", LAMBERT Academic Publishing,
ISBN: 9783847344827 (2012). Thanks to ArXiv for letting us to publish it here
for more acces
The number of dominating -sets of paths, cycles and wheels
We give a shorter proof of the recurrence relation for the domination
polynomial and for the number of
dominating -sets of the path with vertices. For every positive integers
and numbers are determined solving a problem
posed by S. Alikhani in CID 2015. Moreover, the numbers of dominating -sets
of cycles and of wheels with
vertices are computed.Comment: 13 page
Total domination polynomials of graphs
Given a graph , a total dominating set is a vertex set that every
vertex of is adjacent to some vertices of and let be the
number of all total dominating sets with size . The total domination
polynomial, defined as ,
recently has been one of the considerable extended research in the field of
domination theory. In this paper, we obtain the vertex-reduction and
edge-reduction formulas of total domination polynomials. As consequences, we
give the total domination polynomials for paths and cycles. Additionally, we
determine the sharp upper bounds of total domination polynomials for trees and
characterize the corresponding graphs attaining such bounds. Finally, we use
the reduction-formulas to investigate the relations between vertex sets and
total domination polynomials in
On -equivalence classes of some graphs
Let be a simple graph of order . The domination polynomial of is
the polynomial , where is the number
of dominating sets of of size . The -barbell graph with
vertices, is formed by joining two copies of a complete graph by a single
edge. We prove that for every , is not -unique,
that is, there is another non-isomorphic graph with the same domination
polynomial. More precisely, we show that for every , the
-equivalence class of barbell graph, , contains many
graphs, which one of them is the complement of book graph of order ,
. Also we present many families of graphs in
-equivalence class of .Comment: 9 pages, 5 figure
Some new results on the total domination polynomial of a graph
Let be a simple graph of order . The total dominating set of
is a subset of that every vertex of is adjacent to some
vertices of . The total domination number of is equal to minimum
cardinality of total dominating set in and is denoted by . The
total domination polynomial of is the polynomial
, where is the number
of total dominating sets of of size . A root of is called a
total domination root of . An irrelevant edge of is an edge , such that . In this paper, we
characterize edges possessing this property. Also we obtain some results for
the number of total dominating sets of a regular graph. Finally, we study
graphs with exactly two total domination roots , and
.Comment: 12 pages, 7 figure
Final title: "More on domination polynomial and domination root" Previous title: "Graphs with domination roots in the right half-plane"
Let be a simple graph of order n. The domination polynomial of G is the
polynomial D(G,x) =\sum d(G, i)x^i, where d(G,i) is the number of dominating
sets of G of size i. Every root of D(G,x) is called the domination root of G.
It is clear that (0,\infty) is zero free interval for domination polynomial of
a graph. It is interesting to investigate graphs which have complex domination
roots with positive real parts. In this paper, we first investigate complexity
of the domination polynomial at specific points. Then we present and
investigate some families of graphs whose complex domination roots have
positive real part.Comment: 18 Pages, 6 Figures. To appear in Ars Combi
Recurrence relations and splitting formulas for the domination polynomial
The domination polynomial D(G,x) of a graph G is the generating function of
its dominating sets. We prove that D(G,x) satisfies a wide range of reduction
formulas. We show linear recurrence relations for D(G,x) for arbitrary graphs
and for various special cases. We give splitting formulas for D(G,x) based on
articulation vertices, and more generally, on splitting sets of vertices
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