1,064 research outputs found
Operations of graphs and unimodality of independence polynomials
Given two graphs and , assume that is a clique cover of and is a subset of . We introduce a
new graph operation called the clique cover product, denoted by
, as follows: for each clique ,
add a copy of the graph and join every vertex of to every vertex of
. We prove that the independence polynomial of
which
generalizes some known results on independence polynomials of corona and rooted
products of graphs obtained by Gutman and Rosenfeld, respectively. Based on
this formula, we show that the clique cover product of some special graphs
preserves symmetry, unimodality, log-concavity or reality of zeros of
independence polynomials. As applications we derive several known facts in a
unified manner and solve some unimodality conjectures and problems
Domination polynomial of clique cover product of 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 . For two graphs and , let
be a clique cover of and
. We consider clique cover product which denoted by
and obtained from as follows: for each clique
, add a copy of the graph and join every vertex of
to every vertex of . We prove that the domination polynomial of clique
cover product or simply
is
where each clique has vertices. As results, we study the
-equivalence classes of some families of graphs. Also we
completely describe the -equivalence classes of friendship graphs
constructed by coalescence copies of the cycle graph of length three with a
common vertex.Comment: 11 pages, 5 figure
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
On the independent domination polynomial of a graph
An independent dominating set of the simple graph is a vertex
subset that is both dominating and independent in . The independent
domination polynomial of a graph is the polynomial , summed over all independent dominating subsets . A root
of is called an independence domination root. We investigate the
independent domination polynomials of some generalized compound graphs. As
consequences, we construct graphs whose independence domination roots are real.
Also, we consider some certain graphs and study the number of their independent
dominating sets.Comment: 16 pages, 4 figure. arXiv admin note: text overlap with
arXiv:1309.7673 by other author
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
On some conjectures concerning critical independent sets of a graph
Let be a simple graph with vertex set . A set is
independent if no two vertices from are adjacent. For ,
the difference of is and an independent set is
critical if (possibly ). Let and
be the intersection and union, respectively, of all maximum
size critical independent sets in . In this paper, we will give two new
characterizations of K\"{o}nig-Egerv\'{a}ry graphs involving
and . We also prove a related lower bound
for the independence number of a graph. This work answers several conjectures
posed by Jarden, Levit, and Mandrescu.Comment: 10 pages, 3 figure
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
Critical Independent Sets of a Graph
Let be a simple graph with vertex set . A set
is independent if no two vertices from are
adjacent, and by we mean the family of all independent sets
of .
The number is the difference of , and a
set is critical if (Zhang, 1990).
Let us recall the following definitions:
= {S : S is a maximum independent
set}.
= {S :S is a maximum independent
set}.
= {S : S is a critical independent set}.
= {S : S is a critical independent set}.
In this paper we present various structural properties of ,
in relation with , , and .Comment: 15 pages; 12 figures. arXiv admin note: substantial text overlap with
arXiv:1102.113
Critical and Maximum Independent Sets of a Graph
Let G be a simple graph with vertex set V(G). A subset S of V(G) is
independent if no two vertices from S are adjacent. By Ind(G) we mean the
family of all independent sets of G while core(G) and corona(G) denote the
intersection and the union of all maximum independent sets, respectively. The
number d(X)= |X|-|N(X)| is the difference of the set of vertices X, and an
independent set A is critical if d(A)=max{d(I):I belongs to Ind(G)} (Zhang,
1990). Let ker(G) and diadem(G) be the intersection and union, respectively, of
all critical independent sets of G (Levit and Mandrescu, 2012). In this paper,
we present various connections between critical unions and intersections of
maximum independent sets of a graph. These relations give birth to new
characterizations of Koenig-Egervary graphs, some of them involving ker(G),
core(G), corona(G), and diadem(G).Comment: 12 pages, 9 figures. arXiv admin note: substantial text overlap with
arXiv:1407.736
The Independent Domination Polynomial
A vertex subset of the graph is an independent
dominating set if every vertex in is adjacent to at least one
vertex in and the vertices of are pairwise non-adjacent. The
independent domination polynomial is the ordinary generating function for the
number of independent dominating sets in the graph. We investigate in this
paper properties of the independent domination polynomial and some interesting
connections to well known counting problems
- β¦