Connection between Graphs' Chromatic and Ehrhart Polynomials

Journal of Applied Sciences and Nanotechnology, Vol. 3, No. 4 (2023) Journal of Applied Sciences and Nanotechnology Journal homepage: jasn.uotechnology.edu.iq The connection between Graphs' Chromatic and Ehrhart Polynomials 1 Ola A. Neamah, 1Shatha A. Salman* 1Department of Applied Sciences, University of Technology – Iraq ARTICLE INFO ABSTRACT Article history: Received: June, 19, 2023 Accepted: November, 14, 2023 Available online: December, 10, 2023 Graph Theory is a discipline of mathematics with numerous outstanding issues and applications in various sectors of mathematics and science. The chromatic polynomial is a type of polynomial that has valuable and attractive qualities. Ehrhart's polynomials and chromatic analysis are two essential techniques for graph analysis. They both provide insight into the graph's structure but in different ways. The relationship between chromatic and Ehrhart polynomials is an area of active research that has implications for graph theory, combinatorial, and other fields. By understanding the relationship between these two polynomials, one can better understand the structure of graphs and how they interact. This can help us to solve complex problems in our lives more efficiently and effectively. This work gives the relationship between these two essential polynomials and the proof of theorems, and an application related to these works, the model Physical Cell ID (PCID), was discussed. Keywords: Chromatic polynomial, Ehrhart polynomial, h*-polynomial, Symmetric polynomials *Corresponding Author: Shatha A. Salman 100178@uotechnology.edu.iq https://doi.org/10.53293/jasn.2023.6994.1220, Department of Applied Sciences, University of Technology - Iraq. © 2023 The Author(s). This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). 1. Introduction A pair G = (V; E) comprising a set V and a multiset E ⊆ {{i, j}: i, j ∈ V }, is called a graph G. Elements that are vertices or nodes of G make up the set V. The edges of G are in the set E [1] discovered the chromatic polynomial in 1912 while attempting to prove the four-coloring theorem. Since the four-colour theory was first hypothesized in 1852, proving it would have been a significant event. Graph colouring G with n −coloring is a map Ø: V → {1,2, … , n} with dimension n, and the set of colours {1,2, … , n}. A proper n-colouring of the vectors Ø: V → {1,2, … , n} is mapped with the colouring Ø(i) ≠ Ø(j) whenever i and j are adjacent. The number of appropriate graph colourings is counted as a function of n colours by the chromatic polynomial χ(G; n)[2]. Richard Stanley first proposed counting order-preserving maps and its connection to chromatic polynomials in 1970. In other words, given a finite set {v1 , v2 , … , vn } ⊂ ℝd, the smallest convex set containing those points is the polytope P. P is the convex hull of a finite number of points in ℝd . We refer to Po is interior as P. For some k ∈ ℤ>0 and polytope P, kP is a dilated polytope if kP = {(kx1 , kx2 , … , kxd }: (x1 , x2 , … , xd ) ∈ P}. In Ehrhart's theory, the number of lattice points in integral dilates of P is investigated using lattice polytope in ℝd with d dimensions. Fork ∈ ℤ>0 , the lattice point enumerator for the kth dilate is EP (k)= #(kP ∩ ℤd ). The polynomial EP (k) has degree k, with rational coefficients known as the Ehrhart polynomial [3], A lot of applications can be found on [4,5,6 ]. The order polynomial is studied in two areas of mathematics: algebraic graph theory and algebraic combinatorics. The order polynomial counts the number of order-preserving maps from a poset to an ndimensional chain. Richard P. Stanley created these order-preserving maps in 1971 while working on his Ph.D. at 62 O. A. Neamah et al. Journal of Applied Sciences and Nanotechnology, Vol. 3, No. 4 (2023) Harvard University under the direction of Gian-Carlo Rota and researching ordered structures and partitions. In [7] it presents two different geometric explanations for P. Although the order polytope is our primary concern, these two polytopes share some intriguing findings. By maintaining an acyclic orientation, a link may be established between the chromatic polynomials of graph G and the Ehrhart polynomials of the order polytope compatible with various poset in G. The relationship between chromatic polynomials and Ehrhart polynomials has been studied extensively in the field of mathematics, leading to a variety of applications. This relationship has been used to solve problems in graph theory, combinatorics, and geometry, among other areas. [8-10]. It has also provided insight into the properties of convex polytopes and their associated graphs. By understanding the underlying principles behind this relationship, researchers have been able to develop algorithms that can be used to solve various problems related to chromatic polynomials and Ehrhart polynomials. [11-13]. Some of the most important applications of this relationship and how they can be used to solve various real-world problems were discussed. This article defines partially ordered groups and the connection between non-periodic modes and graphs. The reciprocity theorem, and order polynomials Ω(n) are also discussed, in this work, theorems including the connection between order polytope and chromatic polynomials are given. Additionally, the series' numerator polynomial PП is decomposing as, 𝐏 (𝐳) ∑ 𝛀𝐨 (𝐧)𝐳 𝐧 = П 𝐝+𝟏 (𝟏−𝐳) (1) were PП is a polynomial with a nonnegative integer coefficient. 2. Preliminaries This section presents some essential ideas and theorems pertinent to our work. Definition 1[7]: A set П with a binary relation R⊆ П×П that satisfies three requirements is referred to as a partially ordered set or poset.  Reflexivity, if a ∈П then a ≼ a.  Ant symmetry, if a ≼ b and b ≼ a, then a= b  and Transitivity, if a ≼ b and b ≼ c then a ≼ c. Definition 2 [1]: Each pair of vertices in a complete graph is connected by an edge. Definition 3 [7]: An orientation of G is the direction designation by the symbols i⟶j or j⟶i to each edge ij. If there are no coherently directed cycles in an orientation of G, it is acyclic. Fig. 1a, and b shows that K 3 with an acyclic orientation and a noncyclic orientation, respectively. (b) (a) Figure 1: (a) an acyclic orientation of K 3 , (b) a noncyclic orientation. Definition 4[7]: Let П denote a finite poset and n denote an Γn length chain. A chain is an ordered set, specifically the set {1,2,…,n} with the natural order. A map Ø: П → Γn is order preserving if x ≤ y implies that Ø(x) ≤ Ø(y). Let ΩП (n) be the number of such order-preserving maps, and the order polynomial Ω(n)=Ω(p,n) is the function that counts their number. 63 O. A. Neamah et al. Journal of Applied Sciences and Nanotechnology, Vol. 3, No. 4 (2023) Definition 5 [7]: an order polynomial that counts the number of strictly order-preserving maps: П → Γn , meaning x<y implies Ø(x) < Ø(y). Let ΩoП (n) be the number of such strict-order polynomials, then Ωo (n) = Ωo (p, n). Definition 6 [14]: A symmetric polynomial is a polynomial where if it switches any pair of variables, it remains the same. For example, x 2 + y 2 + z2 x 2 + y 2 + z 2 x2 + y 2 + z 2 is a symmetric polynomial, since switching any pair, say x and y, the resulting polynomial y 2 + x 2 + z2 y 2 + x 2 + z 2 y 2 + x 2 + z 2 is the same as the initial polynomial Definition 7 [15]: The set of points (x1 , x2 , … , xn ) in ℝn with 0 ≤ xi ≤ 1 and if t i ≤ p t j then xi ≤ xj is the order polytope 𝒪(П), of a poset П with elements {t1 , t 2 , … , t n }, each point (x1 , x2 , … , xn ) of 𝒪(П) is identified with the function f: П ⟶ ℝ , f(t i ) = xi . Definition 8 [16]: A series of rational numbers known as the Bernoulli numbers B n frequently appears in analysis. The Euler-Maclaurin formula, Faulhaber's formula for the sum of the m-th powers of the first n positive integers, Taylor series expansions of the tangent and hyperbolic tangent functions, and expressions for specific Riemann zeta function values can all be used to define the Bernoulli numbers Bn . The table below contains a list of the first 20 Bernoulli numbers. Table 1: Bernoulli numbers. n 0 Bn 0 1 1 2 2 1 6 3 0 4 −1 30 5 0 6 1 42 7 0 8 1 30 9 0 10 5 66 11 0 12 −691 2730 13 0 14 7 6 15 0 16 3617 510 17 0 18 3867 798 19 0 20 −174611 330 Theorem 1[7]: Order-preserving maps and strictly order-preserving maps are connected that is, (𝐧) = (−𝟏)|П| 𝛀(−𝐧) (2) This restores the negative binomial identity in the case of a chain. The chromatic polynomial and the Ehrhart polynomial are exceptional cases of Stanley's general Reciprocity Theorem yield similar results. Theorem 2[17]: The formulas relate the polynomials ΩП (n) and ΩoП (n) to the Ehrhart polynomial of the order polytope 𝒪П are, 𝐄𝓞П (𝐧) = 𝛀П (𝐧 + 𝟏), 𝐄𝓞𝐨П(𝐧) = 𝛀𝐨П (𝐧 − 𝟏) (3) Theorem 3[7]: The sum of the strict order polynomials for all acyclic orientations σ of a graph G is the chromatic polynomial χn (G). 𝛘𝐧 (𝐆) = ∑𝛔 𝐚𝐜𝐲𝐜𝐥𝐢𝐜 𝛀𝐨П (𝐧) (4) Theorem 4[18]: For any nonnegative integer k, let П be the poset with one minimal element covered by k other elements. It has the Ehrhart polynomial of the order polytope 𝒪(П), 𝐄(𝓞(П), 𝐭) = 𝟏 + ∑𝐤𝐣=𝟏 (𝐁𝐤−𝐣+𝟏 +(𝐤−𝐣+𝟏)) 𝐤 𝐣 (𝐣) 𝐭 𝐤−𝐣+𝟏 𝟏 + 𝐤+𝟏 𝐭 𝐤+𝟏 (5) Were, Bn is the nth Bernoulli number. Theorem 5[18]: For any positive integer j satisfying 1 ≤ j ≤ k − 1, the coefficient of t j of E(𝒪(П), t) is negative if and only if k − j + 1 ≥20 and 4 divides k − j + 1. Hence, the order polytope 𝓞(П) is Ehrhart positive if and only if k < 20. 64 O. A. Neamah et al. Journal of Applied Sciences and Nanotechnology, Vol. 3, No. 4 (2023) 3. Main Results The relationship between chromatic and Ehrhart polynomials has been studied extensively in mathematics. This section will explore the proofs of theorems that link these two polynomials. We will look at how chromatic polynomials can be used to determine Ehrhart polynomials and vice versa. We will also discuss the use cases of these theorems in various contexts. Theorem 6: Let G be a graph with n vertices, χ(G) define the Ehrhart polynomial as E𝒪П , then the relationship between chromatic polynomial and Ehrhart polynomial for graphs is: χn (G) = ∑σ acyclic (−1)|П| E𝒪П (−n − 1) (6) Proof: By using theorem 1 Ωo (n) = (−1)|П| Ω(−n) E𝒪П (n) = ΩП (n + 1) ⟶ E𝒪П (n − 1) = ΩП (n) E𝒪Пo (n) = ΩoП (n − 1). So, E𝒪П (n − 1) = ΩП (n) E𝒪П (−n − 1) = ΩП (−n) (−1)|П| E𝒪П (−n − 1) = (−1)|П| ΩП (−n) = ΩoП (n). Then, (−1)|П| E𝒪П (−n − 1) = ΩoП (n) , χn (G) = ∑σ acyclic ΩoП (n) This is the result. Theorem 7: Let 𝒪(П) be an order polytope of dimension d and let 𝒫П be the numerator polynomial of the series 𝒫 (z) П ∑ Ωo (n)z n = (1−z) d+1 (7) Then 𝒫П can be decomposed as 𝒫П (z) = a(z) + zb(z) where the polynomials a(z) and b(z) are symmetric 1 1 with a(z) = z d a( ) and b(z) = z d−1 ( ) , and so that b(z) and −a(z) have nonnegative coefficients. z z Proof: By using theorem (2) E𝒪П (n) = ΩП (n + 1) (8) And theorem (1) Ωo (n) = (−1)|П| Ω(−n) (9) The following were obtained E𝒪Пo (n) = ΩoП (n − 1) (10) 𝒫 (z) П Put them in ∑ Ωo (n)z n = (1−z) d+1 The results depend on the Bernoulli formula 65 (11) O. A. Neamah et al. Journal of Applied Sciences and Nanotechnology, Vol. 3, No. 4 (2023) E(𝒪(П), t) = 1 + ∑kj=1 (Bk−j+1 +(k−j+1)) k j (j) t k−j+1 1 + k+1 t k+1 (12) According to the relationship between Bernoulli and Eulerian numbers 𝑛 𝐵 𝐸𝑛 = ∑𝑛𝑘=0 ( ) (2𝑘+1 − 22𝑘+2 ) 𝑘+1 ) 𝑘+1 𝑘 (13) k So h∗ (z) = ∑d−1 k=0 E(d, k)z (14) Were the coefficients of the polynomial given by Eulerian numbers that count the number of permutations of {1,…,n} which have k descent., [19]. This means that it gives a symmetric polynomial that satisfying. 1 h∗ (z) = z d−1 h∗ (z) (15) 1 So the numerator 〖 𝒫П (z) is equal z d h∗ (z) =z h∗ (z) , so by decomposition for 𝒫П (z) , a(z)=0 and b(z)= h∗ (z), that satisfies the required conditions. For example, to find the Ehrhart polynomial E(𝒪(П), t) of order polytope 𝒪(П), of dim = k + 1, for k = 4 : (B +(4−j+1)) 1 E(𝒪(П), t) = 1 + ∑4j=1 4−j+1 (16) (41)t j + 4+1 t 4+1 4−j+1 B4 + 4 4 B3 + 3 4 2 B2 + 2 4 3 B1 + 1 4 4 1 4+1 =1+ ( )t+ ( )t + ( )t + ( )t + t 4 1 3 2 2 3 1 4 4+1 1 13 3 1 = 1 − t + 6t 2 + t 3 + t 4 + t 5 3 3 2 5 1 3 Moreover, by direct computation, we see the linear coefficient of E. (𝒪(П), t) equals − t, which is negative. 4. Application for the Ehrhart and Chromatic Polynomials The problem of graph colouring in large, intricate networks like social and informational ones, on the other hand, is a fundamental component of many applications in which the goal is to divide a set of entities into classes where two related entities are not in the same class while also minimizing the number of classes used, despite being useful in a variety of fields (e.g., engineering, scientific computing) [20, 21]. The application, therefore, is given to discuss our idea. 4.1 Numerical Computation Application A femtocell is a small, low-power cellular base station typically designed for home or small business use. Small cell is a broader term that is more widespread in the industry, with femtocell as a subset. The model (PCID) allocation problem with random graph colouring problem. Bernoulli's random graph model is proposed to model deployment scenarios of femtocells, which is applied to allocate PCIDs efficiently, a graph coloring-based PCIDs allocation scheme. Model allocation of PCID as a random Graph colourization problem in femtocellular networks. To solve the problem of PCID allocation with minimal computational expense, the chromatic polynomial of random graphs or order polytopes should be used so that cells overlap femtocells that are coloured differently to avoid a collision. As a result, Figure 2-b depicts each femtocell with its own set of f nodes. An undirected edge connects neighbouring nodes. Second-order neighbours are connected to the same edges to prevent network confusion constraints. The colour will not map until a second-order node is reached, as illustrated in Fig. 2c. 3D shape depicts a scenario in which any algorithm can colourize the graph to avoid collisions and confusion. The number of colours required to colour all of the graph's vertices. 66 O. A. Neamah et al. Journal of Applied Sciences and Nanotechnology, Vol. 3, No. 4 (2023) Figure 2: PCID allocation in femtocellular networks is transformed into a graph colouring algorithm. 5. Conclusions Graph theory has a wide range of areas that depend on the properties of graphs, which are in combinatorial optimization, graph colouring fundamental issues with many applications, such as timetabling and scheduling, frequency assignment, register allocation, and more recently, the analysis of networks of human subjects. The relationship between chromatic and Ehrhart polynomials is meaningful in combinatorial mathematics. This work provided proof of theorems that relate the two polynomials, allowing us to understand better how they are related. We describe how each polynomial's coefficients are described, their respective properties, and how these can be used to prove various results. By understanding these relationships, we can more effectively use chromatic and Ehrhart polynomials to solve problems in combinatorics. More works using this relationship were found in [22]. Acknowledgement The authors thank the University of Technology, Baghdad, Iraq, for technical, instrumental, and infrastructural support during this study. Conflict of Interest The authors declare that they have no conflict of interest. References [1] R.J. Trudeau, "Introduction to graph theory. Courier Corporation", 2013. [2] T.Hubai, and L.László, "The chromatic polynomial", PhD, Eötvös Loránd University, 2009 [3] B.J. 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