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.


Introduction
All graphs in this paper are simple of finite orders, i.e., graphs are undirected with no loops to [24].
Let D(G, i) be the family of dominating sets of a graph G with cardinality i and let d(G, i) = |D(G, i)|. The domination polynomial D(G, x) of G is defined as D(G, x) = |V (G)| i=γ(G) d(G, i)x i (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 D(G, x) is called a domination root of G (see [5,14]). The set of distinct non-zero roots of D(G, x) is denoted by Z * (D(G, x)). It is known that −1 is not a domination root as the number of dominating sets in a graph is always odd [10]. On the other hand, of course, 0 is a domination root of every graph G with multiplicity γ(G). 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 G be the family of graphs and CG = G ∈ G|Z * (D(G, x)) ⊆ C .
In the next section we present some families of graphs whose are in CG. In Section 3 we consider the complement of the friendship graphs, F c n and compute their domination polynomials, exploring the nature and location of their roots. As a consequence we show that F c n ∈ CG.
odd n and the complete bipartite graph K n,n for even n, are in CG. 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 CG?.
In this section we use the existing results on domination polynomials to find some families of graphs whose are in CG. We need some preliminaries.
The join G = G 1 + G 2 of two graphs G 1 and G 2 with disjoint vertex sets V 1 and V 2 and edge sets E 1 and E 2 is the graph union G 1 ∪ G 2 together with all the edges joining V 1 and V 2 . The following theorem gives a formula for the domination polynomial of join of two graphs.

Theorem 1. [2]
Let G and H be nonempty graphs of order n and m, respectively. Then, To present some families of graphs in CG, we recall the existing results.
The book graph B n can be constructed by bonding n copies of the cycle graph C 4 along a common edge {u, v}. In [6] it was proved that, for every n ∈ N, The following theorem gives some families of graphs whose are in CG.

Theorem 3.
(i) [6] Every graph H in the family (ii) [25] For odd n and even k, the k-star S k,n−k is in CG.
(iii) [25] For odd n and odd k, every graph H in the family In [28], Levit and Mandrescu constructed a family of graphs H n from the path P n by the "clique cover construction", as shown in Figure 1. By H 0 we mean the null graph. Figure 1: Graphs H 2n+1 and H 2n , respectively.
The following theorem gives formula for the domination polynomials of H n graphs: Let H n be the graphs in the Figure 1.
Here using Theorem 4 we present another families of graphs in CG.
(i) The graphs of the form H n + H n , H n+1 + B n , for n ≥ 3, and the graphs of the form B n + B n , for odd n are in CG.
(ii) The graphs of the form B n+1 + B n , for even n, and B n+1 + H n , for n ≥ 4 are in CG.
Proof. Since the coefficients of domination polynomials are positive integers, we investigate domination roots for x ≤ 0.
(i) By theorem 1 we can deduce that for each natural number n ≥ 3, To obtain the domination roots of H n + H n , we shall solve the following equation: We consider two cases, and show that in each there is no nonzero solution.
• If n ≥ 3 is even, i.e., n = 2k for some k ∈ N. Then the equation (1) is equivalent to the following equation For x ≤ 0, the above equality is true just for real number 0. Because for nonzero real number the left side of this equality is positive but the right side is negative.
• If n ≥ 3 is odd, i.e., n = 2k + 1, n = 2k for some k ∈ N. Then the equation (1) is is equivalent to the following equation We consider the following different cases, and show in each there is no nonzero solution. If x ≤ −1, there are no real solutions x. Because, it is easy to see that for −2 ≤ x ≤ −1, the left side of 3 is positive but its right side is negative. Also for x < −2, the left side of equality (3) is greater than the right side. Now suppose that (a) If k is even and − 1 2 ≤ x < 0, the left side of equality (3) is greater than the right side, a contradiction.
(b) If k is odd and − 1 2 ≤ x < 0, the left side of equality (3) is positive but the right side is negative, a contradiction.

Domination roots of the complement of the friendship graphs
The friendship (or Dutch-Windmill) graph F n is a graph that can be constructed by coalescence n copies of the cycle graph C 3 of length 3 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 n, the friendship graph F n lie in CG.
(ii) For odd n, F n ∈ CG.
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 T (n, r) is a complete multipartite graph formed by partitioning a set of n vertices into r subsets, with sizes as equal as possible, and connecting two vertices by an edge whenever they belong to different subsets. The graph will have (n mod r) subsets of size ⌈ n r ⌉, and r − (n mod r) subsets of size ⌊ n r ⌋. That is, it is a complete r-partite graph The Turán graph T (2n, n) can be formed by removing a perfect matching, n edges no two of which are adjacent, from a complete graph K 2n . As Roberts (1969) showed, this graph has boxicity exactly n; it is sometimes known as the Roberts graph [29]. If n 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 It is easy to check that the complement of the friendship graph F n is CP (n) ∪ K 1 . Figure 4 shows the complement of the friendship graph F n .
Proof. An elementary observation is that if G 1 and G 2 are graphs of orders n 1 and n 2 , respectively, then Clearly D(K 1 , x) = x and there are no dominating sets of size 1 in CP (n). Therefore 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 F c n ) 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 n, the complement of the friendship graphs F c n lie in CG.
Theorem 8. For every natural number n, the complement of the friendship graphs F c n lie in CG.
Proof. It's suffices to show that for each natural n, the cocktail party graph CP (n) is in CG. We consider three cases, and show in each there is no nonzero solution.
• x > 0 : Obviously the above equality is true just for real number 0, since for nonzero real number the left side of equality is greater than the right side.
• x ≤ −1 : In this case the left side is greater than 0 and the right side 1 + 2nx is less than −1, a contradiction.
• −1 < x < 0 : In this case obviously there are no real solutions x, 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 f n (x) is a family of (complex) polynomials, we say that a number z ∈ C is a limit of roots of f n (x) if either f n (z) = 0 for all sufficiently large n or z is a limit point of the set R(f n (x)), where R(f n (x)) is the union of the roots of the f n (x).
The following restatement of the Beraha-Kahane-Weiss theorem [9] can be found in [12].
Theorem 9. Suppose f n (x) is a family of polynomials such that where the α i (x) and the λ i (x) are fixed non-zero polynomials, such that for no pair i = j is λ i (x) ≡ ωλ j (x) for some ω ∈ C of unit modulus. Then z ∈ C is a limit of roots of f n (x) if and only if either (i) two or more of the λ i (z) are of equal modulus, and strictly greater (in modulus) than the others; or (ii) for some j, λ j (z) has modulus strictly greater than all the other λ i (z), and α j (z) = 0 The following Theorem gives the limits of the domination roots of F c n .
Theorem 10. The limit of domination roots of F c n is the unit circle with center −1.
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.