Cookies policy. I'm starting in Wolfram Mathematica. graph J Egypt Math Soc 27, 18 (2019). Furthermore, \(\chi(C_n) \ne 2\) since vertex colors cannot alternate, as the final vertex to be colored will be adjacent to both a red and a blue vertex. A coloring using at most n colors is called n-coloring. Empty graphs have chromatic number 1, while non-empty Math. Case 2. Should I trust my own thoughts when studying philosophy? All https://mathworld.wolfram.com/MinimumVertexColoring.html. Why did my papers got repeatedly put on the last day and the last session of a conference? Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. How can I conveniently call igraph from Mathematica? The chromatic number of a surface of genus is given by the Heawood This is by far the fastest implementation that exists for Mathematica, and is competitive with other systems. to be weakly perfect. It only takes a minute to sign up. There are various steps to solve the greedy algorithm, which are described as follows: Step 1: In the first step, we will color the first vertex with first color. If you look at a tree, for instance, you can obviously color it in two colors, but not in one color, which means a tree has the chromatic number 2. Thus, for the most part, one must be content with supplying bounds for the chromatic number of graphs. Proof. The chromatic number of a graph is the smallest number of colors needed to color the vertices Vorici Chromatic Calculator Note: Chromatic orbs cannot reroll the same color permutation twice, so the chromatic success chance is always higher than the drop rate. Does the Earth experience air resistance? where 1 ≤ l ≤ m, 1 ≤ j ≤ m and 1 ≤ s ≤ n. Any column l has the entry \( {a}_{2l}^{\ast }=f \) put, Again repeat steps (c) and (d) to complete the RF coloring matrix. Math. A vertex coloring that minimize the number of colors needed for a given In the following, we give examples solved by the new algorithm: Example1. By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. $$, $$ \mathrm{RCI}(G)=\left[\begin{array}{c}1\\ {}1\\ {}\begin{array}{c}0\\ {}0\\ {}\begin{array}{c}0\\ {}0\end{array}\end{array}\end{array}\kern0.5em \begin{array}{c}0\\ {}3\\ {}\begin{array}{c}3\\ {}0\\ {}\begin{array}{c}0\\ {}0\end{array}\end{array}\end{array}\kern0.5em \begin{array}{ccc}\begin{array}{c}2\\ {}0\\ {}\begin{array}{c}2\\ {}0\\ {}\begin{array}{c}0\\ {}0\end{array}\end{array}\end{array}& \begin{array}{c}0\\ {}2\\ {}\begin{array}{c}0\\ {}2\\ {}\begin{array}{c}0\\ {}0\end{array}\end{array}\end{array}& \begin{array}{ccc}\begin{array}{c}3\\ {}0\\ {}\begin{array}{c}0\\ {}3\\ {}\begin{array}{c}0\\ {}0\end{array}\end{array}\end{array}& \begin{array}{c}4\\ {}0\\ {}\begin{array}{c}0\\ {}0\\ {}\begin{array}{c}0\\ {}4\end{array}\end{array}\end{array}& \begin{array}{ccc}\begin{array}{c}0\\ {}0\\ {}\begin{array}{c}1\\ {}0\\ {}\begin{array}{c}0\\ {}1\end{array}\end{array}\end{array}& \begin{array}{c}0\\ {}0\\ {}\begin{array}{c}5\\ {}0\\ {}\begin{array}{c}5\\ {}0\end{array}\end{array}\end{array}& \begin{array}{ccc}\begin{array}{c}0\\ {}4\\ {}\begin{array}{c}0\\ {}0\\ {}\begin{array}{c}4\\ {}0\end{array}\end{array}\end{array}& \begin{array}{c}0\\ {}0\\ {}\begin{array}{c}0\\ {}1\\ {}\begin{array}{c}1\\ {}0\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}0\\ {}0\\ {}\begin{array}{c}0\\ {}5\\ {}\begin{array}{c}0\\ {}5\end{array}\end{array}\end{array}& \begin{array}{c}0\\ {}0\\ {}\begin{array}{c}0\\ {}0\\ {}\begin{array}{c}2\\ {}2\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right]. Is it just the way it is we do not say: consider to do something? If we want to color a graph with the help of a minimum number of colors, for this, there is no efficient algorithm. MathJax reference. $$, $$ {a}_{11}^{\ast }=\kern0.75em \left\{\begin{array}{c}0\kern0.5em \mathrm{if}\kern0.75em {a}_{11}=0\\ {}\ \\ {}1\ \mathrm{if}\kern0.75em {a}_{11}=1\end{array}\right.. $$, $$ {a}_{12}^{\ast }=\left\{\begin{array}{ccc}0& \mathrm{if}& {a}_{12}=0\ \\ {}1& \mathrm{if}& {a}_{12}=1,{a}_{11}=0\ \\ {}2& \mathrm{if}& {a}_{11}={a}_{12}=1\end{array}\right.. $$, $$ {a}_{1j}^{\ast }=\kern0.75em \left\{\begin{array}{c}\kern4.25em 0\kern10.50em \mathrm{if}\kern0.75em {a}_{1j}=0\\ {}\ \\ {}\max \left\{{a}_{11}^{\ast },{a}_{12}^{\ast },\dots, {a}_{1\left(j-1\right)}^{\ast}\right\}+1\kern1em \mathrm{if}\kern0.75em {a}_{1j}=1\end{array}\ \right.. $$, $$ {a}_{ik}^{\ast }=\kern0.75em \left\{\begin{array}{c}h\kern1em \mathrm{if}\kern0.75em {a}_{ik}=1\\ {}\ \\ {}0\kern1em \mathrm{if}\kern0.5em {a}_{ik}=0\end{array}\ \right.. $$, $$ \left\{\begin{array}{ccc}{a}_{2l}^{\ast }=0& \mathrm{if}& {a}_{2l}=0\\ {}\min \left(\left\{{a}_{1j}^{\ast }:{a}_{1j}^{\ast}\ne 0,\right\}\backslash \left\{{a}_{2j}^{\ast },{a}_{sj}^{\ast }:{a}_{2j}^{\ast}\ne 0,{a}_{sj}^{\ast}\ne 0\right\}\right)& \mathrm{if}& {a}_{2l}=1,{a}_{Sl}=1,\left\{{a}_{1j}^{\ast }:{a}_{1j}^{\ast}\ne 0\right\}\backslash \left\{{a}_{2j}^{\ast },{a}_{Sj}^{\ast }:{a}_{2j}^{\ast}\ne 0\ne {a}_{Sj}^{\ast}\right\}\ne \AE \\ {}\max \left(\left\{{a}_{1j}^{\ast },{a}_{2j}^{\ast },{a}_{sj}^{\ast }:{a}_{1j}^{\ast}\ne 0,{a}_{2j}^{\ast}\ne 0\ne {a}_{sj}^{\ast },\right\}\right)+1& \mathrm{if}& {a}_{2l}=1,{a}_{Sl}=1,\left\{{a}_{1j}^{\ast }:{a}_{1j}^{\ast}\ne 0\right\}\backslash \left\{{a}_{2j}^{\ast },{a}_{Sj}^{\ast }:{a}_{2j}^{\ast}\ne 0\ne {a}_{Sj}^{\ast}\right\}=\AE \end{array}\right. Sign up, Existing user? Minimal colorings and chromatic numbers for a sample of graphs are illustrated above. Many day-to-day problems, like minimizing conflicts in scheduling, are also equivalent to graph colorings. Chromatic number = 2. Let's take a tree with n ( ≥ 2) vertices as an example. Click two nodes in turn to add an edge between them. The IGraph/M package has an implementation of this. Browse other questions tagged. Suppose there are \(n\) colors among the vertices from \(G\), and suppose there are \(m\) colors among the vertices from \(G'\). if G=(V,E), is a connected graph and e belong E . Math. For any two positive integers and , there exists a graph of girth at least and chromatic number at least (Erdős 1961; Lovász 1968; Skiena 1990, p. 215). What is the minimal number \(k\) such that there exists a proper edge coloring of the complete graph on 8 vertices with \(k\) colors? Get machine learning and engineering subjects on your finger tip. It only takes a minute to sign up. By Observation 4, I [ r] also has independence number three and thus its independence ratio is 3 12 r = 1 4 r. Since the chromatic number is at least the inverse of independence ratio, the chromatic number of the r -inflation is at least 4 r. "no convenient method is known for determining the chromatic number of an arbitrary I was wondering if there is a way to calculate the chromatic number of a graph knowing only the chromatic polynomial, but not the actual graph. Chromatic number can be described as a minimum number of colors required to properly color any graph. https://brilliant.org/wiki/graph-coloring-and-chromatic-numbers/. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Grzesik and Khachatrian [2] proved that k1, m, n is interval colorable iff gcd(m + 1, n + 1) = 1. The original article was written in Japanese here. In the above graph, we are required minimum 3 numbers of colors to color the graph. For mono-requirement items, on-color: 0.9 * (R + 10) / (R + 20) For mono-requirement items, off-color: 0.05 + 4.5 / (R + 20) For dual-requirement items, on-color: 0.9 * R1 / (R1 + R2) I am so grateful to the reviewers for their many valuable suggestions and comments that significantly improved the paper. Let H be a subgraph of G. Then (G) ≥ (H). Hence the chromatic number K n = n. Mahesh Parahar. is said to be a k-chromatic graph. Chromatic number of a graph is the minimum value of k for which the graph is k - c o l o r a b l e. In other words, it is the minimum number of colors needed for a proper-coloring of the graph. The author worked on the results and also read and approved the final manuscript. So. is known as a minimum vertex coloring of . and a graph with chromatic number is said to be three-colorable. The degree of \(P_G\) is equal to the number of vertices of \(G\). "ChromaticNumber"]. RF coloring algorithm is introduced as follows: Consider a graph G of order n and size m. List its vertices as v1, v2, v3 , … , vn and its edges as e1, e2, e3 , … , em. Hence, each vertex requires a new color. Use MathJax to format equations. The converse statement is an easier problem to approach: are all graphs with chromatic number at most four planar? Prove that, \[\chi(C_n) = \begin{cases} 2 & \text{if } n \text{ is even} \\ 3 & \text{if } n \text{ is odd.} Springer-Verlag, Berlin Heidelberg (2007), Lehner, F.: Breaking graph symmetries by edge colourings. Theorem 1 Let Sn be a star graph of order n. Then, χ ΄ (Sn) = n − 1, where χ ΄ (Sn) is the chromatic index of Sn. Now we want to evaluate the values of l and m and we have two cases: Case 1. Another type of edge coloring is used in Ramsey theory and similar problems. Example: << IGraphM` g = RandomGraph [ {10, 20}] Compute the chromatic number: IGChromaticNumber [g] (* 4 *) Compute a minimum colouring: $$ \chi_G = \min \{k \in \mathbb N ~|~ P_G(k) > 0 \} $$. Chromatic number is the minimum number of colors to color all the vertices, so that no two adjacent vertices have the same color. It has a great scope for further research in the field of graph theory, computer programming, and algebraic-specific structures. n = |V (G)| = |V1| |V2| ... |Vk| ≤ k (G) = (G) (G). We observe that there exist { P 5, K 4 } -free graphs with chromatic number equal to 5. How can I speed up the classic GA for graph coloring? When n is even then l is the entry in \( {a}_{\left(n-1\right)\left(n-2\right)}^{\ast } \), i.e., l = 2. https://mathworld.wolfram.com/ChromaticNumber.html. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. We can improve a “best possible” bound by obtaining another bound that is always at least as good. The best answers are voted up and rise to the top, Not the answer you're looking for? hz abbreviation in "7,5 t hz Gesamtmasse". 5, 174–176 (2000) Elsevier, Kandel, A., Bunke, H., Last, M.: Applied Graph Theory in Computer Vision and Pattern Recognition. conjecture. (Thanks to Juho for the guidance on this!). bipartite graphs have chromatic number 2. Generalizing the construction of K n, m, the graph join of two graphs is the graph obtained by joining (with . rev 2023.6.5.43477. 2014, 7 (2014) Hindawi Publishing Corporation. The 4-coloring of the graph G shown in Figure 3.2 establishes that (G) ≤ 4, and the K4-subgraph (drawn in bold) shows that (G) ≥ 4. $$, $$ \mathrm{RCI}\left({S}_n\right)=\left[\begin{array}{ccc}1& 2\kern0.5em 3& \kern0.5em \begin{array}{cc}\dots & n-1\end{array}\\ {}\begin{array}{c}1\\ {}0\end{array}& \begin{array}{c}\begin{array}{cc}0& 0\end{array}\\ {}\begin{array}{cc}2& 0\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\dots \kern1em \\ {}\dots \kern1.25em \end{array}& \begin{array}{c}0\\ {}0\end{array}\end{array}\\ {}\begin{array}{c}\vdots \\ {}0\end{array}& \begin{array}{c}\begin{array}{cc}\vdots & \vdots \end{array}\\ {}\begin{array}{cc}0& 0\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\kern0.5em \ddots \\ {}\dots \end{array}& \begin{array}{c}\kern0.75em \vdots \\ {}n-1\end{array}\end{array}\end{array}\right] $$, \( \chi \prime \left({C}_n\right)=\left\{\begin{array}{c}2\kern1em if\kern0.75em n\ is\ even\\ {}\ \\ {}3\kern1em if\kern1em n\ is\ odd\end{array}\right. A graph coloring is an assignment of labels, called colors, to the vertices of a graph such that no two adjacent vertices share the same color. Notes Discret. Wolfram. For , 1, ..., the first few values of are 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, Other types of colorings on graphs also exist, most notably edge colorings that may be subject to various constraints. The independence polynomial is given by (2) and the matching polynomial by (3) (4) Graphing Calculator Loading. Then u001f (G) ≥ k. Proof. Is a quantity calculated from observables, observable? Proposition 2. \(_\square\), Suppose a graph \(G\) and a graph \(G'\) are combined to create a graph \(H\) by connecting each vertex of \(G\) to each vertex of \(G'\) and otherwise all vertices and edges remaining unchanged. https://mat.tepper.cmu.edu/trick/color.pdf. Chi-boundedness and Upperbounds on Chromatic Number. @Szabolcs Ah, thanks! Movie with a scene where a robot hunter (I think) tells another person during dinner that you can recognize a cyborg by the creases in their fingers, Lilypond: \downbow and \upbow don't show up in 2nd staff tablature. and chromatic number (Bollobás and West 2000). A few basic principles recur in many chromatic-number calculations. Could algae and biomimicry create a carbon neutral jetpack? Creative Commons Attribution 4.0 International License. Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Graph coloring is also known as the NP-complete algorithm. We will color the currently picked vertex with the help of lowest number color if and only if the same color is not used to color any of its adjacent vertices. polynomial . Electron. Finding a minimal Solution. But a graph coloring for \(C_n\) exists where \(n - 1\) vertices are alternately colored red and blue and the final vertex is colored yellow, so \(\chi(C_n) = 3\). The chromatic number of a graph must be greater than or equal to its clique number. To learn more, see our tips on writing great answers. Can expect make sure a certain log does not appear? List of the Chromatic Polynomial formulas with simple graphs # When graph have 0 edge # Where n is the number of Vertices. By definition, the edge chromatic number of a graph The chromatic polynomial χ G ( t) of a graph G = ( V, E) can always be written as. Most upper bounds on the chromatic number come from algorithms that produce colorings. The bound Δ(G) 1 is the worst upper bound that greedy coloring could produce. 127, 205–214 (2017) Elsevier, Article (You can simplify your computations by thinking about the effect on the chromatic polynomial of deleting an edge that is a loop, or deleting . Why is this screw on the wing of DASH-8 Q400 sticking out, is it safe? Therefore, we can say that the Chromatic number of above graph = 3. Let V be the set of vertices of a graph. In a complete graph, each vertex is adjacent to is remaining (n–1) vertices. Let G be a graph with k-mutually adjacent vertices. Since \( {a}_{\left(n-1\right)\left(n-2\right)}^{\ast }=2 \) and \( {a}_{nm}^{\ast }=2 \) then m = 1; hence, the RF coloring matrix is. Connect and share knowledge within a single location that is structured and easy to search. Then, the neighbors of each of those vertices also has \(k-1\) possible colors, and so on. 1, 123–132 (2011), MATH Let (G) be the independence number of G, we have Vi ≤ (G). This graph is 4-colorable. Definition 1. From the wikipedia page for Chromatic Polynomials: The chromatic polynomial includes at least as much information about the colorability of G as does the chromatic number. Prove that \(\chi(G) + \chi(G') = \chi(H).\). Basic Principles for Calculating Chromatic Numbers: Although the chromatic number is one of the most studied parameters in graph theory, no formula exists for the chromatic number of an arbitrary graph. Therefore, we can say that the Chromatic number of above graph = 4. Hence, (G) = 4. The most common type of edge coloring is analogous to graph (vertex) colorings. Corollary 1. is the floor function. The above discussions motivate us to design a new algorithm to calculate the chromatic index of the graph. Further, Lehner [6] established a result state that if every non-trivial automorphism of a countable graph G with distinguishing index D ΄ (G) moves infinitely many edges, then D ΄ (G) ≤ 2. Because every vertex in \(G\) is adjacent to every vertex in \(G'\), the two vertex sets cannot have any color in common. Proof Let Sn be a star graph of order n as shown in Fig. FIGURE 2.22. Google Scholar, Luna, G., Romero, J.R.M., Moyao, Y.: An approximate algorithm for the chromatic number of graphs. Does a knockout punch always carry the risk of killing the receiver? If you can divide all the vertices into K independent sets, you can color them in K colors because no two adjacent vertices share the edge in an independent set. It is based on encoding the colouring problem into a Boolean satisfiability problem. Mathematica package to calculate the graph crossing number, How to determine that a plane graph is an Apollonian network, Currency Converter (calling an api in c#). For any subsets , let me define ind(U) as 'the number of subsets of U, which compose an independent set.'. Sudoku can be seen as a graph coloring problem, where the squares of the grid are vertices and the numbers are colors that must be different if in the same row, column, or \(3 \times 3\) grid (such vertices in the graph are connected by an edge). Determine the chromatic number of each connected graph. Mathematica Stack Exchange is a question and answer site for users of Wolfram Mathematica. Terms and Conditions, Math. i.e., the smallest value of possible to obtain a k-coloring. http://mathworld.wolfram.com/MinimumVertexColoring.html, Chromatic number for "great circle" graph, What developers with ADHD want you to know, MosaicML: Deep learning models for sale, all shapes and sizes (Ep. It is impossible to color the graph with 2 colors, so the graph has chromatic number 3. Computation of the chromatic number of a graph is implemented in the Wolfram Language as VertexChromaticNumber[g]. A graph has a chromatic number that is at most one larger than the chromatic number of a subgraph containing only one less vertex. Chromatic number of a graph G is denoted by χ ( G). The smallest number of colors needed to color a graph G is called its chromatic number, and is often denoted ch. Explore math with our beautiful, free online graphing calculator. Sign up to read all wikis and quizzes in math, science, and engineering topics. Python Code: def chromatic_polynomial (lambda, vertices): return lambda ** vertices Complete Graph # By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. $$, \( {a}_{\left(n-1\right)\left(n-2\right)}^{\ast } \), \( {a}_{\left(n-1\right)\left(n-2\right)}^{\ast }=2 \), $$ \mathrm{RCI}\left({C}_n\right)=\left[\begin{array}{c}\begin{array}{c}\begin{array}{cc}1& 0\end{array}\kern0.5em \begin{array}{cc}0& \begin{array}{cc}\dots & \begin{array}{cc}0& \begin{array}{cc}0& 2\end{array}\end{array}\end{array}\end{array}\\ {}\begin{array}{cc}\begin{array}{cc}1& 2\end{array}& \begin{array}{cc}0& \begin{array}{cc}\dots & \begin{array}{cc}0& \begin{array}{cc}0& 0\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\\ {}\begin{array}{c}\begin{array}{cc}\begin{array}{cc}0& 2\end{array}& \begin{array}{cc}1& \begin{array}{cc}\dots & \begin{array}{cc}0& \begin{array}{cc}0& 0\end{array}\end{array}\end{array}\end{array}\end{array}\\ {}\begin{array}{c}\begin{array}{cc}\begin{array}{cc}\vdots & \vdots \kern0.5em \end{array}& \begin{array}{cc}\vdots & \begin{array}{cc}\ddots & \begin{array}{cc}\vdots &\ \begin{array}{cc}\vdots & \vdots \end{array}\end{array}\end{array}\end{array}\end{array}\\ {}\begin{array}{c}\begin{array}{cc}\begin{array}{cc}0& 0\end{array}& \begin{array}{cc}0& \begin{array}{cc}\dots & \begin{array}{cc}2& \begin{array}{cc}0& 0\end{array}\end{array}\end{array}\end{array}\end{array}\\ {}\begin{array}{c}\begin{array}{cc}\begin{array}{cc}0& 0\end{array}& \begin{array}{cc}0& \begin{array}{cc}\dots & \begin{array}{cc}2& \begin{array}{cc}1& 0\end{array}\end{array}\end{array}\end{array}\end{array}\\ {}\begin{array}{cc}\begin{array}{cc}0& 0\end{array}& \begin{array}{cc}0& \begin{array}{cc}\dots & \begin{array}{cc}0& \begin{array}{cc}1& 2\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right] $$, \( {a}_{\left(n-1\right)\left(n-2\right)}^{\ast }=1 \), $$ \mathrm{RCI}\left({C}_n\right)=\left[\begin{array}{c}\begin{array}{c}\begin{array}{cc}1& 0\end{array}\kern0.5em \begin{array}{cc}0& \begin{array}{cc}\dots & \begin{array}{cc}0& \begin{array}{cc}0& 2\end{array}\end{array}\end{array}\end{array}\\ {}\begin{array}{cc}\begin{array}{cc}1& 2\end{array}& \begin{array}{cc}0& \begin{array}{cc}\dots & \begin{array}{cc}0& \begin{array}{cc}0& 0\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\\ {}\begin{array}{c}\begin{array}{cc}\begin{array}{cc}0& 2\end{array}& \begin{array}{cc}1& \begin{array}{cc}\dots & \begin{array}{cc}0& \begin{array}{cc}0& 0\end{array}\end{array}\end{array}\end{array}\end{array}\\ {}\begin{array}{c}\begin{array}{cc}\begin{array}{cc}\vdots & \vdots \kern0.5em \end{array}& \begin{array}{cc}\vdots & \begin{array}{cc}\ddots & \begin{array}{cc}\vdots &\ \begin{array}{cc}\vdots & \vdots \end{array}\end{array}\end{array}\end{array}\end{array}\\ {}\begin{array}{c}\begin{array}{cc}\begin{array}{cc}0& 0\end{array}& \begin{array}{cc}0& \begin{array}{cc}\dots & \begin{array}{cc}1& \begin{array}{cc}0& 0\end{array}\end{array}\end{array}\end{array}\end{array}\\ {}\begin{array}{c}\begin{array}{cc}\begin{array}{cc}0& 0\end{array}& \begin{array}{cc}0& \begin{array}{cc}\dots & \begin{array}{cc}1& \begin{array}{cc}3& 0\end{array}\end{array}\end{array}\end{array}\end{array}\\ {}\begin{array}{cc}\begin{array}{cc}0& 0\end{array}& \begin{array}{cc}0& \begin{array}{cc}\dots & \begin{array}{cc}0& \begin{array}{cc}3& 2\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right], $$, $$ \left[\begin{array}{ccc}\begin{array}{c}\begin{array}{c}1\\ {}0\\ {}0\end{array}\\ {}\begin{array}{c}0\\ {}1\\ {}0\end{array}\\ {}\begin{array}{c}0\\ {}0\\ {}0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}2\\ {}0\\ {}0\end{array}\\ {}\begin{array}{c}0\\ {}0\\ {}2\end{array}\\ {}\begin{array}{c}0\\ {}0\\ {}0\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}3\\ {}0\\ {}0\end{array}\\ {}\begin{array}{c}0\\ {}0\\ {}0\end{array}\\ {}\begin{array}{c}3\\ {}0\\ {}0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ {}2\\ {}0\end{array}\\ {}\begin{array}{c}0\\ {}2\\ {}0\end{array}\\ {}\begin{array}{c}0\\ {}0\\ {}0\end{array}\end{array}\end{array}\end{array}\kern0.5em \begin{array}{ccc}\begin{array}{c}\begin{array}{c}0\\ {}3\\ {}0\end{array}\\ {}\begin{array}{c}0\\ {}3\\ {}0\end{array}\\ {}\begin{array}{c}0\\ {}0\\ {}0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ {}1\\ {}0\end{array}\\ {}\begin{array}{c}0\\ {}0\\ {}1\end{array}\\ {}\begin{array}{c}0\\ {}0\\ {}0\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ {}0\\ {}3\end{array}\\ {}\begin{array}{c}0\\ {}0\\ {}3\end{array}\\ {}\begin{array}{c}0\\ {}0\\ {}0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ {}0\\ {}1\end{array}\\ {}\begin{array}{c}0\\ {}0\\ {}0\end{array}\\ {}\begin{array}{c}0\\ {}1\\ {}0\end{array}\end{array}\end{array}\end{array}\kern0.5em \begin{array}{ccc}\begin{array}{c}\begin{array}{c}0\\ {}0\\ {}2\end{array}\\ {}\begin{array}{c}0\\ {}0\\ {}0\end{array}\\ {}\begin{array}{c}0\\ {}0\\ {}2\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ {}0\\ {}0\end{array}\\ {}\begin{array}{c}1\\ {}0\\ {}0\end{array}\\ {}\begin{array}{c}1\\ {}0\\ {}0\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}\begin{array}{c}0\\ {}0\\ {}0\end{array}\\ {}\begin{array}{c}2\\ {}0\\ {}0\end{array}\\ {}\begin{array}{c}2\\ {}0\\ {}0\end{array}\end{array}& \begin{array}{c}\begin{array}{c}0\\ {}0\\ {}0\end{array}\\ {}\begin{array}{c}3\\ {}0\\ {}0\end{array}\\ {}\begin{array}{c}0\\ {}0\\ {}3\end{array}\end{array}\end{array}\end{array}\right] $$, https://doi.org/10.1186/s42787-019-0018-9, Journal of the Egyptian Mathematical Society, http://creativecommons.org/licenses/by/4.0/.
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