Algebraic Integers as Chromatic and Domination Roots

Let G be a simple graph of order n and λ ∈ N. A mapping f : V G → {1, 2, . . . , λ} is called a λ-colouring of G if f u / f v whenever the vertices u and v are adjacent in G. The number of distinct λ-colourings of G, denoted by P G, λ , is called the chromatic polynomial of G. The domination polynomial of G is the polynomial D G, λ ∑n i 1 d G, i λ , where d G, i is the number of dominating sets of G of size i. Every root of P G, λ and D G, λ is called the chromatic root and the domination root of G, respectively. Since chromatic polynomial and domination polynomial are monic polynomial with integer coefficients, its zeros are algebraic integers. This naturally raises the question: which algebraic integers can occur as zeros of chromatic and domination polynomials? In this paper, we state some properties of this kind of algebraic integers.


Introduction
Let G be a simple graph and λ ∈ N. A mapping f : V G → {1, 2, . . ., λ} is called a λ-colouring of G if f u / f v whenever the vertices u and v are adjacent in G.The number of distinct λ-colourings of G, denoted by P G, λ , is called the chromatic polynomial of G.A zero of P G, λ is called a chromatic zero of G.For a complete survey on chromatic polynomial and chromatic root, see 1 .
For any vertex v ∈ V , the open neighborhood of v is the set N v {u ∈ V | uv ∈ E}, and the closed neighborhood is the set N v N v ∪ {v}.For a set S ⊆ V , the open neighborhood of S is N S v∈S N v , and the closed neighborhood of S is N S N S ∪ S. A set S ⊆ V is a dominating set if N S V , or, equivalently, every vertex in V \ S is adjacent to at least one vertex in S.An i-subset of V G is a subset of V G of cardinality i.Let D G, i be the family of dominating sets of G which are i-subsets and let d G, i |D G, i |.The polynomial D G, x We denote the set of all roots of D G, x by Z D G, x .For more information and motivation of domination polynomial and domination roots, refer to 2-6 .
We recall that a complex number ζ is called an algebraic number respectively, algebraic integer if it is a zero of some monic polynomial with rational resp., integer coefficients see 7 .Corresponding to any algebraic number ζ, there is a unique monic polynomial p with rational coefficients, called the minimal polynomial of ζ over the rationals , with the property that p divides every polynomial with rational coefficients having ζ as a zero.The minimal polynomial of ζ has integer coefficients if and only if ζ is an algebraic integer.
Since the chromatic polynomial and domination polynomial are monic polynomial with integer coefficients, its zeros are algebraic integers.This naturally raises the question: which algebraic integers can occur as zeros of chromatic and domination polynomials?
In Sections 2 and 3, we study algebraic integers as chromatic roots and domination roots, respectively.
As usual, we denote the complete graph of order n and the complement of G, by K n and G, respectively.

Algebraic Integers as Chromatic Roots
Since chromatic polynomial is monic polynomial with integer coefficients, its zeros are algebraic integers.An interval is called a zero-free interval for a chromatic domination polynomial, if G has no chromatic domination zero in this interval.It is well known that −∞, 0 and 0, 1 are two maximal zero-free intervals for chromatic polynomials of the family of all graphs see 8 .Jackson 8 showed that 1, 32/27 is another maximal zerofree interval for chromatic polynomials of the family of all graphs and the value 32/27 is best possible.
For chromatic polynomials clearly those roots lying in −∞, 0 ∪ 0, 1 ∪ 1, 32/27 are forbidden.Tutte  Also they extended this result to show that φ 2n and all their natural powers cannot be chromatic zeros, where φ n is called n-annaci constant 12 .
For some times it was thought that chromatic roots must have nonnegative real part.This is true for graphs with fewer than ten vertices.But Sokal showed the following.Theorem 2.2 see 13 .Complex chromatic roots are dense in the complex plane.

Theorem 2.3. The set of chromatic roots of a graph G is not a semiring.
Proof.The set of chromatic roots is not closed under either addition or multiplication, because it suffices to consider α α * and αα * , where α is nonreal and close to the origin.Theorem 2.4.Suppose that a, b are rational numbers, r ≥ 2 is an integer that is not a perfect square, and a − |b| √ r < 32/27.Then a b √ r is not the root of any chromatic polynomial.
Proof.If λ a b √ r is a root of some polynomial with integer coefficients e.g., a chromatic polynomial , then so is λ * a − b √ r.But λ or λ * cannot belong to −1, 0 ∪ 0, 1 ∪ 1, 32/27 , a contradiction.We know that for every graph G with edge e xy, P G, λ P G e, λ P G • e, λ , where G • e is the graph obtained from G by contracting x and y and removing any loop.By applying this recursive formula repeatedly, we arrive at where b i 's are some constants and Let us recall the definition of join of two graphs.The join of two graphs G 1 and G 2 , denoted by Theorem 2.6 see 14 .Let G 1 and G 2 be any two graphs with P G i , λ expressed in factorial form, i 1, 2. Then where ⊗ is called umbral product, and acts as powers (i.e., λ i ⊗ λ j λ i j .
Here we state and prove the following theorem.
Theorem 2.7.For any graph H, Proof.It suffices to prove it for n 1. Assume that P H, λ i≥1 b i λ i .By Theorem 2.6,

2.5
Here we state and prove the following theorem.
Theorem 2.8.If α is a chromatic root, then for any natural number n, α n is a chromatic root. Proof.Since we have the result.
By Theorem 2.1, τ and τ 1 τ 2 are not chromatic roots.However τ 3 is a chromatic root see Theorem 2.14 .Therefore by Theorem 2.8 we have the following corollary.
Corollary 2.9.For every natural number n ≥ 3, τ n is a chromatic root.
There are the following conjectures.
Conjecture 2.10 see 15 .Let α be an algebraic integer.Then there exists a natural number n such that α n is a chromatic root.Conjecture 2.11 see 15 .Let α be a chromatic root.Then nα is a chromatic root for any natural number n. Definition 2.12 see 15 .A ring of cliques is the graph R a 1 , . . ., a n whose vertex set is the union of n 1 complete subgraphs of sizes 1, a 1 , . . ., a n , where the vertices of each clique are joined to those of the cliques immediately preceding or following it mod n 1. Theorem 2.13 see 15 .The chromatic polynomial of R a 1 , . . ., a n is a product of linear factors and the polynomial

2.7
We call the polynomial in Theorem 2.13 the interesting factor.
Remark 2.15.We observed that τ n is a chromatic root for every n ≥ 3. Also we saw that τ 1 is not a chromatic root, but we do not know whether τ 2 is a chromatic root or not.Therefore this remains as an open problem.

Algebraic Integers as Domination Roots
For domination polynomial of a graph, it is clear that 0, ∞ is zero-free interval.Brouwer 16 has shown that −1 cannot be domination root of any graph G.For more details of the domination polynomial of a graph at −1 refer to 17 .We also have shown that every integer domination root is even 18 .
Let us recall the corona of two graphs.The corona of two graphs G 1 and G 2 , as defined by Frucht and Harary in 19 , is the graph G G 1 • G 2 formed from one copy of G 1 and |V G 1 | copies of G 2 , where the ith vertex of G 1 is adjacent to every vertex in the ith copy of G 2 .The corona G • K 1 , in particular, is the graph constructed from a copy of G, where for each vertex v ∈ V G , a new vertex v and a pendant edge vv are added.
Here we state the following theorem.
Theorem 3.1 see 2 .Let G be a graph.Then D G, x x n x 2 n if and only if G H • K 1 for some graph H of order n.
By above theorem there are infinite classes of graphs which have −2 as domination roots.Since −1 is not domination root of any graph, so we do not have result for domination roots similar to Theorem 2.8.Also we think that the following conjecture is correct.Conjecture 3.2 see 18 .If r is an integer domination root of a graph, then r 0 or r −2.Now we recall the following theorem.
The following corollary is an immediate consequence of above theorem.The following theorem state that −τ cannot be a domination root.Here we will prove that −τ n for odd n, cannot be a domination root.We need some theorems.
Theorem 3.8 see 20 .For every natural number n,

3.1
Corollary 3.9.For every natural number n Proof.This follows from Theorem 3.8.Now, we recall the Cassini's formula.
Using this formula, we prove another property of golden ratio and Fibonacci numbers which is needed for the proof of Theorem 3.13.Theorem 3.11.

3.4
Proof.Suppose that n is even, therefore n − 1 is odd, and by Corollary 3.9, we have Hence, and by multiplying F n in this inequality, we have 3.8 By Theorem 3.10, we have 3.9 Hence, for even n,

3.10
Similarly, the result holds when n is odd.

Corollary 2 . 5 .
Let b be a rational number, and let r be a positive rational number such that √ r is irrational.Then b √ r cannot ba a root of any chromatic polynomial.

Theorem 3 . 5 . 5 / 2 >Theorem 3 . 6 .Corollary 3 . 7 .
−τ cannot be a domination root.Proof.Let G be any graph.Since D G, −τ is a polynomial with integral coefficients, we have D G, 0, a contradiction.The following theorem is similar to Theorem 3.6 for domination roots.Suppose that a, b are rational numbers, r ≥ 2 is an integer that is not a perfect square, and a − |b| √ r < 0. Then −a − b √ r is not the root of any domination polynomial.Proof.If λ −a−b √ r is a root of some polynomial with integer coefficients e.g., a domination polynomial , then so is λ * −a b √ r.But λ * ∈ 0, ∞ , a contradiction.Let b be a rational number, and let r be a positive rational number such that √ r is irrational.Then −|b| √ r cannot ba a root of any domination polynomial.
Theorem 3.12 see 20, page 78 .For every n ≥ 2,τ n F n τ F n−1 n ≥ 2 .Now we are ready to prove the following theorem.Let n be an odd natural number.Then −τ n cannot be domination roots.