Note on chromatic polynomials of the threshold graphs

Let G be a threshold graph. In this paper, we give, in first hand, a formula relating the chromatic polynomial of G (the complement of G) to the chromatic polynomial of G. In second hand, we express the chromatic polynomials of G and G in terms of the generalized Bell polynomials.


Introduction
Recall that for a given graph G = (V, E) of order n, a λ-coloring of G, λ ∈ N, is a mapping f : V → {1, 2, . . ., λ} where f (u) ̸ = f (v) whenever the edge uv ∈ E. If such mapping f exists, the graph G is said to be λ-colorable, the chromatic number of G, denoted by χ (G) , is the minimal value of λ for which the graph G is λ-colorable and the number of λ-colorings of G is called the chromatic polynomial P (G, λ), see [4,8,9].This paper is concerned with the chromatic polynomials and the sigma polynomials of threshold graphs.These graphs was introduced by Chvátal et al. [3] and Henderson et al. [5] and have numerous applications, see for example [6].They can be constructed from an isolated vertex by repeated applications to addition a vertex to be an isolated vertex or a dominating vertex to the graph.From this definition, it follows that the complement graph G of G is also a threshold graph.The object of our investigations in this paper is, in first hand, to deduce for a given threshold graph G the chromatic polynomial P (G, λ) from the chromatic polynomial P (G, λ), and, in second hand to express the chromatic polynomials of G and G in terms of the generalized Bell polynomials B r,s (x) defined by Carlitz and studied extensively by Blasiak, Penson and Solomon, see [1,2].Below, we use the following notation: G n is a graph of order n and without edges with the convention P (G 0 , λ) = 1.

Chromatic polynomials of threshold graphs
Upon using the definition of threshold graphs G and G, the following theorem gives simple expressions for their chromatic polynomials.
Theorem 2.1.Let (G n , n ≥ 1) be a sequence of threshold graphs and G n has n vertices.Then Furthermore, we have where with i 0 = j 0 = 0 and δ is the Kronecker delta, i.e. δ (i,j) = 1 if i = j and δ (i,j) = 0 if i ̸ = j.
Proof.By construction, G n is the graph G n−1 plus a vertex x n such that x n is an isolated vertex or a dominating vertex.Similarly, by construction, G n is the graph which can be written as Thus, the desired expressions follow.Let now r k be the order of multiplicity of a number k of the zeros of P (G n , λ) : For a given threshold graph G with known chromatic polynomial P (G, λ) , the following theorem gives the explicit expression of the chromatic polynomial Theorem 2.2.Let (G n ; n ≥ 1) be a sequence of threshold graphs and G n has n vertices such that for some non-negative integers r 0 , r 1 , . . ., r n−1 such that Then, the following holds Proof.From Theorem 2.1, the chromatic polynomial P (G n , λ) can be written asP We prove that the chromatic polynomial P ( G n , λ ) must be as follows Indeed, by induction on n.The case n = 1 is obvious and assume where where s 0 = 1, s j = r j−1 (1 ≤ j ≤ n) , and since λ = λ − s 0 + 1 we get So, the induction is true and produces the desired result.
Corollary 2.1.Let G be a threshold graph of n vertices.Then Proof.It is easy to see that Corollary 2.2.Let G be a threshold graph of n vertices.Then, the sum of all zeros of the polynomial Proof.Setting for some non-negative integers r 0 , r 1 , . . ., r n−1 and s 0 , s 1 , . . ., s n−1 such that is the sum of all zeros of P (G, λ) (resp.P ( G, λ ) ), then, from Theorem 2.2 the sum of all zeros of the polynomial

The generalized Bell polynomials and threshold graphs
To give some connections between the chromatic polynomials and the generalized Bell polynomials (see [7]), let r 0 , . . ., r n−1 and s 0 , . . ., s n−1 be non-negative integers and set r = (r 0 , . . ., r n−1 ), s = (s 0 , . . ., s n−1 ) .Recall that the generalized Stirling numbers of the second kind S r,s (n, k) are defined by and the so-called generalized Bell polynomials B r,s (x) are to be By choosing f (x) = x λ in the identity we obtain [7].
For a given sequence (G n , n ≥ 1) of threshold graphs with G n has n vertices, we prove in this section that the sequence of the sigma polynomials (σ(G n , x), n ≥ 1) can be expressed in terms of the generalized Bell polynomials.The useful representation of the chromatic polynomial of a given graph G = (V, E) used here is where |V | is the number of vertices of V and α i (G) is the number of ways of partitioning V into i nonempty sets.The sigma polynomial σ(G, λ) of a graph G = (V, E) is defined by Proof.From the definition of the chromatic polynomial of G we get exp (−x) The following theorem shows that some generalized Bell polynomials can be interpreted by the chromatic polynomials for the threshold graphs and gives another version of Theorem 2.2.