1 Introduction

Some families of graphs with no nonzero real domination roots

S. Jahari111Corresponding author. E-mail: s.jahari@gmail.com and S. Alikhani

Department of Mathematics, Yazd University

89195-741, Yazd, Iran

ABSTRACT

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 . A root of is called a domination root of . Obviously, is a domination root of every graph with multiplicity . 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.

Mathematics Subject Classification: 05C31, 05C60.
Keywords: Domination polynomial; domination root; friendship; complex root.

## 1 Introduction

All graphs in this paper are simple of finite orders, i.e., graphs are undirected with no loops or parallel edges and with finite number of vertices. Let be a simple graph. For any vertex , the open neighborhood of is the set and the closed neighborhood of is the set . For a set , the open neighborhood of is and the closed neighborhood of is .

The complement of a graph is a graph with the same vertex set as and with the property that two vertices are adjacent in if and only if they are not adjacent in .

A set is a dominating set if or equivalently, every vertex in is adjacent to at least one vertex in . The domination number is the minimum cardinality of a dominating set in . For a detailed treatment of domination theory, the reader is referred to [24].

Let be the family of dominating sets of a graph with cardinality and let . The domination polynomial of is defined as (see [2, 8, 27]); this polynomial is the generating polynomial for the number of dominating sets of each cardinality. Similar to generating polynomials for other combinatorial sequences, such as independents sets in a graph [11, 13, 15, 18, 21, 22, 23], have attracted recent attention, to name but a few. The algebraic encoding of salient counting sequences allows one to not only develop formulas more easily, but also, often, to prove unimodality results via the nature of the the roots of the associated polynomials (a well known result of Newton states that if a real polynomial with positive coefficients has all real roots, then the coefficients form a unimodal sequence (see, for example, [16]). A root of is called a domination root of (see [5, 14]). The set of distinct non-zero roots of is denoted by . It is known that is not a domination root as the number of dominating sets in a graph is always odd [10]. On the other hand, of course, is a domination root of every graph with multiplicity . The existing research on the roots of domination polynomials has been restricted to those graphs with exactly two, three or exactly four domination roots ([2, 3]). Also in [14] Brown and Tufts studied the location of the roots of domination polynomials for some families of graphs such as bipartite cocktail party graphs and complete bipartite graphs. In particular, they showed that the set of all domination roots is dense in the complex plane.

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 would like to present some families of graphs with this property. Let be the family of graphs and .

In the next section we present some families of graphs whose are in . In Section we consider the complement of the friendship graphs, and compute their domination polynomials, exploring the nature and location of their roots. As a consequence we show that .

## 2 Some families of graphs in CG

In the beginning of the study of domination roots of graphs, one can see that there are graphs with no nonzero real domination roots except zero. As examples, the complete graph for odd and the complete bipartite graph for even , are in . With these motivation, in [1, 5] the authors asked the question: “Which graphs have no nonzero real domination roots?” In other words, which graph lie in ?.

In this section we use the existing results on domination polynomials to find some families of graphs whose are in . We need some preliminaries.

The join of two graphs and with disjoint vertex sets and and edge sets and is the graph union together with all the edges joining and . The following theorem gives a formula for the domination polynomial of join of two graphs.

###### Theorem 1

.[2] Let and be nonempty graphs of order and , respectively. Then,

 D(G+H,x)=((1+x)n−1)((1+x)m−1)+D(G,x)+D(H,x).

For two graphs and , the corona is the graph arising from the disjoint union of with copies of , by adding edges between the th vertex of and all vertices of th copy of [19]. We need the following theorem which is for computing the domination polynomial of the corona products of two graphs.

###### Theorem 2

.[4, 26] Let and be nonempty graphs of order and , respectively. Then

 D(G∘H,x)=(x(1+x)m+D(H,x))n.

To present some families of graphs in , we recall the existing results.

A -star, , has vertex set where and for .

The book graph can be constructed by bonding copies of the cycle graph along a common edge . In [6] it was proved that, for every ,

 D(Bn,x)=(x2+2x)n(2x+1)+x2(x+1)2n−2xn.

The following theorem gives some families of graphs whose are in .

###### Theorem 3

.

1. [6] Every graph in the family lie in .

2. [25] For odd and even , the -star is in .

3. [25] For odd and odd , every graph in the family

 {G∘Sk,n−k,(G∘Sk,n−k)∘Sk,n−k,((G∘Sk,n−k)∘Sk,n−k)∘Sk,n−k,⋯}

lie in .

4. [6] Every graph in the family lie in .

In [28], Levit and Mandrescu constructed a family of graphs from the path by the “clique cover construction”, as shown in Figure 1. By we mean the null graph.

The following theorem gives formula for the domination polynomials of graphs:

###### Theorem 4

.[7] Let be the graphs in the Figure 1.

1. For every ,

2. For every ,

Here using Theorem 4 we present another families of graphs in .

###### Theorem 5

.

1. The graphs of the form , , for , and the graphs of the form , for odd are in .

2. The graphs of the form , for even , and , for are in .

Proof. Since the coefficients of domination polynomials are positive integers, we investigate domination roots for .

1. By theorem 1 we can deduce that for each natural number ,

 D(Hn+Hn,x)=((1+x)|V(Hn)|−1)2+2D(Hn,x).

To obtain the domination roots of , we shall solve the following equation:

 ((1+x)|V(Hn)|−1)2=−2D(Hn,x). (1)

We consider two cases, and show that in each there is no nonzero solution.

• If is even, i.e., for some . Then the equation (1) is equivalent to the following equation

 ((1+x)4k−1)2 = −2(x4+4x3+6x2+2x)k (2) = −2((1+x)4−2x−1)k.

For , the above equality is true just for real number . Because for nonzero real number the left side of this equality is positive but the right side is negative.

• If is odd, i.e., , for some . Then the equation (1) is is equivalent to the following equation

 ((1+x)4k+3−1)2 = −2(x3+3x2+x)(x4+4x3+6x2+2x)k (3) = −2((1+x)3−2x−1)((1+x)4−2x−1)k.

We consider the following different cases, and show in each there is no nonzero solution. If , there are no real solutions . Because, it is easy to see that for , the left side of 3 is positive but its right side is negative. Also for , the left side of equality (3) is greater than the right side. Now suppose that .

1. If is even and the left side of equality (3) is greater than the right side, a contradiction.

2. If is odd and the left side of equality (3) is positive but the right side is negative, a contradiction.

3. For every and there are no real solutions . Because the left side of equality (3) is positive but the right side is negative.

The other cases are similar to this case.

2. It is similar to proof of Part (i).

Domination roots of the graphs , for odd and  has shown in Figure 2.

## 3 Domination roots of the complement of the friendship graphs

The friendship (or Dutch-Windmill) graph is a graph that can be constructed by coalescence copies of the cycle graph of length with a common vertex. The Friendship Theorem of Paul Erdös, Alfred Rényi and Vera T. Sós [17], states that graphs with the property that every two vertices have exactly one neighbour in common are exactly the friendship graphs. Figure 3 shows some examples of friendship graphs.

The following theorem states that for each odd , the friendship graph lie in .

###### Theorem 6

.[6]

1. For every ,

2. For odd , .

Domination polynomials, exploring the nature and location of domination roots of friendship graphs has studied in [6]. It is natural to ask about the domination polynomial and the domination roots of the complement of the friendship graphs.

The Turán graph is a complete multipartite graph formed by partitioning a set of vertices into subsets, with sizes as equal as possible, and connecting two vertices by an edge whenever they belong to different subsets. The graph will have subsets of size , and subsets of size . That is, it is a complete -partite graph

 K⌈nr⌉,⌈nr⌉,…,⌊nr⌋,⌊nr⌋.

The Turán graph can be formed by removing a perfect matching, edges no two of which are adjacent, from a complete graph . As Roberts (1969) showed, this graph has boxicity exactly ; it is sometimes known as the Roberts graph [29]. If couples go to a party, and each person shakes hands with every person except his or her partner, then this graph describes the set of handshakes that take place; for this reason it is also called the cocktail party graph. So, the cocktail party graph of order is the graph with vertices in which each pair of distinct vertices form an edge with the exception of the pairs .

It is easy to check that the complement of the friendship graph is . Figure 4 shows the complement of the friendship graph .

###### Theorem 7

. For every ,

Proof. An elementary observation is that if and are graphs of orders and , respectively, then

 D(G1∪G2,x)=D(G1,x)D(G2,x).

Clearly and there are no dominating sets of size in . Therefore

 D(CP(n),x)=(1+x)2n−(1+2nx).\vbox{\hrule width 1% 00% height 1px\hbox{\vrule width 1px\kern 0.0pt\vbox{\kern 0.0pt\vbox to 5.690% 551pt{\hfill}\kern 0.0pt}\kern 0.0pt\vrule width 1px}\hrule width 100% height % 1px}\vskip12.0ptplus4.0ptminus4.0pt

In [14] a family of graphs was produced with roots just barely in the right-half plane (showing that not all domination polynomials are stable), but Figure 5 provides an explicit family (namely the ) whose domination roots have positive real part.

The domination roots of complement of the friendship graphs exhibit a number of interesting properties (see Figure 5). Even though we cannot find the roots explicitly, there is much we can say about them.

Here we prove that for each natural number , the complement of the friendship graphs lie in .

###### Theorem 8

. For every natural number , the complement of the friendship graphs lie in .

Proof. It’s suffices to show that for each natural , the cocktail party graph is in . By Theorem 7, for every , If then for , we have

 (1+x)2n=1+2nx.

We consider three cases, and show in each there is no nonzero solution.

• Obviously the above equality is true just for real number , since for nonzero real number the left side of equality is greater than the right side.

• In this case the left side is greater than and the right side is less than , a contradiction.

• In this case obviously there are no real solutions , the left side of equality is greater than the right side.

Thus in any event, there are no nonzero real domination roots of the cocktail party graph.

The plot in Figure 5 suggests that the roots tend to lie on a curve. In order to find the limiting curve, we will need a definition and a well known result.

###### Definition 1

. If is a family of (complex) polynomials, we say that a number is a limit of roots of if either for all sufficiently large or z is a limit point of the set , where is the union of the roots of the .

The following restatement of the Beraha-Kahane-Weiss theorem [9] can be found in [12].

###### Theorem 9

. Suppose is a family of polynomials such that

 fn(x)=α1(x)λ1(x)n+α2(x)λ2(x)n+...+αk(x)λk(x)n (4)

where the and the are fixed non-zero polynomials, such that for no pair is for some of unit modulus. Then is a limit of roots of if and only if either

• two or more of the are of equal modulus, and strictly greater (in modulus) than the others; or

• for some , has modulus strictly greater than all the other , and

The following Theorem gives the limits of the domination roots of .

###### Theorem 10

. The limit of domination roots of is the unit circle with center .

Proof. By Theorem 7, the domination polynomial of is,

 D(Fcn,x) = x((x+1)2)n−x(1+2nx) = α1(x)λn1(x)+α2(x)λn2(x),

where

 α1(x)=x,  λ1(x)=(x+1)2,

and

 α2(x)=x+2nx2,  λ2(x)=1.

Clearly there is no of modulus for which (or vice versa). Also, are not identically zero. Therefore, the initial conditions of Theorem 9 are satisfied. Now, implies that lies on the circle centred at .

Conclusion. In this paper we presented some families of graphs whose non-zero domination roots are complex. We think that these kind of graphs shall have specific geometrical properties. However, until now all attempts to find these properties failed, and it remains as open problem.

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