Formal Calculation uses an auxiliary form to calculate various nested sums and provides results in three forms. It is also a powerful tool for analysis. This article studies the symmetry of the coefficients in Formal Calculation. Three types of extended numbers were introduced, and many formulas for symmetric functions were obtained.

Formal Calculation uses an auxiliary form to calculate various nested sums and provides results in three forms. In addition to computation, it is also a powerful tool for analysis, allowing one to study various numbers in a unified way. This article contains many results of two types of Stirling numbers, associated Stirling numbers, and Eulerian numbers, making a great generalization of Euler polynomials, Wilson's theorem, and Wolstenholme's theorem, showing that they are just special cases. Formal Calculation provides a novel method for obtaining combinatorial identities and analyzing q-binomial.This article has obtained many results in q-analogues, including inversion formulas for q-binomial coefficients. This article also introduces a theorem on symmetry

Chord diagrams are cubic graphs with two types of edges: the first set of edges comprise a subgraph which is a simple cycle (the frame); the second type of edges (the chords) comprise disconnected 2-vertex subgraphs incident to distinct vertices of the frame. We define associated cubic graphs with directed chords (arcs) while keeping the edges of the frame undirected, and plot all 1, 3, 13, and 121 of them for 2, 4, 6, and 8 vertices.

We show that the Graph Isomorphism (GI) problem can be solved in polynomialO(n3 + n2 m log(n + m) + m2 n log(n + m) + m3 log(n + m)) time, on simple con-nected graphs with n vertices and m edges. As fundamental part of the proof,we introduce a novel method, named Symmetry Classication, which computesa canonical automorphism partition of a simple connected graph in polynomialO(n2 + nm log n) time. The master algorithm reduces to bipartite graph iso-morphism by transforming the input graph, if not bipartite, to its subdivisiongraph and computes the canonical form of the latter; or of the former if alreadybipartite. Canonical form is obtained by repeating the sequence of rst applyingSymmetry Classication method, then canonically selecting a vertex for indi-vidualization, and last applying the classical 1-dimensional Weisfeiler-Lehmanalgorithm, until a canonical discrete coloring is attained. Since both bipartiteand connected graph isomorphism are isomorphism complete we conclude thatGI lies in P.

Using ideas from the geometry of compression, we improve on the current upper bound of Heilbronn's triangle problem. In particular, by letting $Delta(s)$ denotes the minimal area of the triangle induced by $s$ points in a unit disc, then we have the upper bound $$Delta(s)ll frac{1}{s^{frac{3}{2}-epsilon}}$$ for small $epsilon:=epsilon(s)>0$.

This is an introduction to graph theory, from a geometric viewpoint. A finite graph $X$ is described by its adjacency matrix $din M_N(0,1)$, which can be thought of as a kind of discrete Laplacian, and we first discuss the basics of graph theory, by using $d$ and linear algebra tools. Then we discuss the computation of the classical and quantum symmetry groups $G(X)subset G^+(X)$, which must leave invariant the eigenspaces of $d$. Finally, we discuss similar questions for the quantum graphs, with these being again described by certain matrices $din M_N(mathbb C)$, but in a more twisted way.

A wazir is a fairy chess piece that attacks the 4 neighbors to the North, East, South and West of the chess board. This work constructs the bivariate generating functions for the number of placing w mutually non-attacking wazirs on rectangular boards of shape r X c at fixed c. The equinumerous setup counts binary {0,1}-arrays of dimension r X c which have w 1's with mutual L1 (Manhattan) distances > 1.

The problem of determining whether a graph has a fixed-point-free automorphism is NP-complete. We demonstrate that the problem can be solved efficiently within polynomial time. First, we obtain the automorphisms of an input graph G by using a spectral method. Next, we prove the Theorem used to detect whether there is a fixed-point free automorphism in G. Next, we construct an algorithm to detect whether G has a fixed-point-free automorphism using this result. The computational complexity of detecting whether a graph has a fixed-point-free automorphism is O(n^5). If fixed-point-free automorphism exist, the computational complexity of obtaining a fixed-point-free automorphism is O(n^6). Then, the complexity classes P and NP are the same.

We show that the graph isomorphism problem is solvable in polynomial time. First, we prove the theorem to obtain the automorphisms of G using eigenvalue sets. Next, we construct an algorithm to determine whether two given graphs G_a and G_b are isomorphic using this result. The computational complexity to detect whether the two graphs are isomorphic is O(n^6).

Vertex cover problem is a famous combinatorial problem, which its complexity has been heavily studied over the years and while a 2-approximation for it can be trivially obtained, researchers have not been able to approximate it better than 2-o(1). In this paper, by a combination of a new semidefinite programming formulation along with satisfying new proposed properties, we introduce an approximation algorithm for the vertex cover problem with a performance ratio of 1.999999 on arbitrary graphs, en route to answering an open question about the correctness/incorrectness of the unique games conjecture.

In this paper, we have explained how logic-vectors can be interpreted as a geometrical representation over computational engines and how it can be implemented in code using large language models. We have also proposed the usage of Time Compass to derive logic-vector evolution over a quasi-chaotic system and its direct manipulation on the tessellation Colormap. This paper focuses on the optimal arrangement of reflecting points for efficient ray tracing given limited sweep time. We examine spatial configurations, employing our core concept of a sweeping subnet and defining a causal barrier to capture constraints imposed by time.We will also discuss the influence of these constructions on the design of an algorithm for approximating optimal tessellations.I have provided code for each of the graphs, as the mathematics is demonstrated unequivocally by their implementation. The reader can test out the reality of this system by visualizing the graphs themselves using Python in a suitable environment like Google Colaboratory.

We introduce eight infinite sets of constants.Some we calculate. Roughly speaking, we seek graphs as small as possible. The graphs serve as examples for different kinds of 'dimensions'. In the second part 'Points', we place points on the plane, and from this, we ask infinite many questions. In a third part, we introduce for each graph a sequence called the 'Thuerey Numbers'.At the end, we summarize the open questions.

Hypergraphs are generalization of graphs, which have several useful applications. Sunflower hypergraphs are interesting hypergraphs, which become linear in some cases. In this paper, we discuss the Siedel spectrum of these hypergraphs.

This paper gives an explicit isomorphic mapping from the 240 real R^8 roots of the E8 Gossett 4_{21} 8-polytope to two golden ratio scaled copies of the 120 root H4 600-cell quaternion 4-polytope using a traceless 8x8 rotation matrix U with palindromic characteristic coefficients and a unitary form e^{iU}}. It also shows the inverse map from a single H4 600-cell to E8 using a 4D<->8D chiral L<->R mapping function, phi scaling, and U^{-1}. This approach shows that there are actually four copies of each 600-cell living within E8 in the form of chiral H4L+phi H4L+H4R+phi H4R roots. In addition, it demonstrates a quaternion Weyl orbit construction of H4-based 4-polytopes that provides an explicit mapping between E8 and four copies of the tri-rectified Coxeter-Dynkin diagram of H4, namely the 120-cell of order 600. Taking advantage of this property promises to open the door to as yet unexplored E8-based Grand Unified Theories or GUTs.

The Harmonic Graphs Conjecture states that there exists an asymptotic relation involving the Harmonic Index and the natural logarithm as the order of the graph increases. This conjecture, grounded in the novel context of Prime Graphs, draws upon the Prime Number Theorem and the sum of divisors function to unveil a compelling asymptotic connection. By carefully expanding the definitions of the harmonic index and the sum of divisors function, and leveraging the prime number theorem's approximations, we establish a formula that captures this intricate relationship. This work is an effort to contribute to the advancement of graph theory, introducing a fresh lens through which graph connectivity can be explored. The synthesis of prime numbers and graph properties not only deepens our understanding of structural complexity but also paves the way for innovative research directions.

We introduce a general conjecture involving minimum-link covering trails for any given k-dimensional grid n × n × ··· × n, belonging to the cubic lattice ℕ^k. In detail, if n is above two, we hypothesize that the minimal link length of any covering trail, for the above-mentioned set of n^k points in the Euclidean space ℝ^k, is equal to h(n, k) = (n^k − 1)/(n − 1) + c·(n − 3), where c = k − 1 iff h(4, 3) = 23, c = 1 iff h(4, 3) = 22, or even c = 0 iff h(4, 3) = 21.

A multiset of u*b balls contains u different colors and b balls of each color. Randomly distributing them across u urns with b balls per urn, what is the probability that no urn contains at least two balls of a common color? We reduce this problem to the enumeration of u X u binary matrices with constant row and column sum b and provide an explicit table of probabilities for small b and u.

In this note, we give an alternate proof of the multinomial theorem following the number $ A_n^p $ using probabilistic approach. Although the multinomial theorem following the number $ A_n^p$ is basically a combinatorial result, our proof may be simple for a student familiar with only basic probability concepts.

In this paper we obtain the multinomial theorem following the numbers $ A_n^p $ and $ C_n^p $ (Vandermonde's identity generalization). Using this notion we obtain generalization of products of numbers in arithmetic progression, arithmetic regression and their sum. From the generalisation we propose (define) the arithmetics sequences product.

The article deals with labeled rooted growing trees. Research in this area, carried outby the author of this article over the past 35 years, has led to the creation of the conceptof tree classification of labeled graphs. This concept is the mathematical basis of the treesum method aimed at simplifying the representations of the coefficients of power series in classical statistical mechanics. This method was used to obtain tree representations of Mayer coefficients of expansions of pressure and density in terms of activity degrees, which are free from asymptotic catastrophe. The same method was used to obtain tree representations of thecoefficients of the expansion of the ratio of activity to density in terms of activity degrees.All these representations for n ≥ 7 are much simpler than the comparable Ree-Hooverrepresentations according to the complexity criteria defined on these representations. Treerepresentations of the coefficients of the expansion of the m-particles distribution functioninto a series in terms of activity degrees were also obtained. All the above representations ofthe coefficients of power series obtained by the trees sum method are free from the asymptoticcatastrophe.In order to provide a mathematical basis for constructing new, even less complexrepresentations of the coefficients of these power series, further development of the conceptof tree classification of labeled graphs was required.As part of solving the problem of further development of this concept, the article proposesnew classifications of labeled rooted growing trees. And on their basis, the theorem wasformulated and proved, which is the basis for simplifying the tree representations of functions,that is, its representations as a sum of labeled by trees products of functions.

In this paper, we provide an elementary proof of Franel number recurrence relation. Consequently, we derive a recurrence relation involving the sum of third powers of binomial coefficients.

Using the method of compression we show that the number of integral points in the annular region induced by two $k$ dimensional spheres of radii $r$ and $R$ with $R>r$ satisfies the lower bound \begin{align} \mathcal{N}_{R,r,k} \gg (R^{k-1}-r^{k+\delta})\sqrt{k}.\nonumber \end{align}for some small $\delta>0$ with $k>\frac{\delta(\log r)}{\log R-\log r}$.

Inspired by Mark Levi's wonderful proof of the Cosine addition formula, that showed that it follows from the sad fact that Perpetual Motion is impossible, we recall five other proofs.

Bridge graphs are special type of graphs which are constructed by connecting identical connected graphs with path graphs. We discuss different type of bridge graphs $B_{n\times l}^{m\times k}$ in this paper. In particular, we discuss the following: complete-type bridge graphs, star-type bridge graphs, and full binary tree bridge graphs. We also bound the second eigenvalues of the graph Laplacian of these graphs using methods from Spectral Graph Theory. In general, we prove that for general bridge graphs, $B_{n\times l}^2$, the second eigenvalue of the graph Laplacian should between $0$ and $2$, inclusive. At the end, we talk about the future work about infinite bridge graphs. We create definitions and found the related theorems to support our future work about infinite bridge graphs.

We prove the veracity of the Syracuse conjecture by establishing that starting from an arbitrary positive integer diffrent of $1$ and $4$, the Syracuse process will never comeback to any positive integer reached before and then we conclude by using a probabilistic approach.

In this paper we continue with the development of the circles of partitions by introducing the notion of the area induced by circles of partitions and explore some applications.

The thinness of a simple graph G= (V,E) is the smallest integer k for which there exist a total order (V, <) and a partition of V into k classes (V_1,...,V_k) such that, for all u, v, w in V with u<v<w, if u and v belong to the same class and {u,w} is in E, then {v,w} is in E. We prove that (1) there are $n$-vertex graphs of thinness $n-o(n)$, which answers a question of Bonomo-Braberman, Gonzalez, Oliveira, Sampaio, and Szwarcfiter, (2) the computation of thinness is NP-hard, which is a solution to a long standing open problem posed by Mannino and Oriolo.

The Collatz Conjecture remains an intriguing problem. Expanding on an altered formula first presented in “Bits of Complexity”, this paper explores the concept that “3N + Least-Significant-Bit” of a number allows the complete separation of the step of “dividing by two on an even number” within the Collatz Conjecture. This alternate formula replaces the original Collatz on a one-for-one basis. The breadth and depth of graphs of the resulting path of numbers resolving with this new formula illustrate its fractal nature. Lastly, we explore the predictability of this data, and how the ultimate goal of reaching one prevented previous work from finding the key to understanding 3N+1.

In this paper, we combine a real or complex palindromic sequence and arithmetic sequence to produce the sums of product of powers of palindromic-arithmetic sequences. As a result, we generate new expressions for Franel numbers as well as first Strehl identity.

In this paper, we present a problem concerning the sum of powers of Binomial coefficients. We prove two special cases of the problem using some simple identities involving Binomial coefficients, and list another two cases but without proof.

The partial automorphisms of a graph $X$ having $N$ vertices are the bijections $\sigma:I\to J$ with $I,J\subset\{1,\ldots,N\}$ which leave invariant the edges. These bijections form a semigroup $\widetilde{G}(X)$, which contains the automorphism group $G(X)$. We discuss here the quantum analogue of this construction, with a definition and basic theory for the quantum semigroup of quantum partial automorphisms $\widetilde{G}^+(X)$, which contains both $G(X)$, and the quantum automorphism group $G^+(X)$. We comment as well on the case $N=\infty$, which is of particular interest, due to the fact that $\widetilde{G}^+(X)$ is well-defined, while its subgroup $G^+(X)$, not necessarily, at least with the currently known methods.

This paper explores populating adjacency matrices with connected cycles whose final outputs represent the coefficients of rational generating functions (RGFs). An RGF takes the form of: $p(x)/q(x) + r(x)$. The denominator, $q(x)$, takes the form of: Constant $\cdot (1-c_1x^{x_1})(1-c_2x^{x_2})... (1-c_nx^{x_n})$ where the $c_i$ are complex numbers and where factors can possibly have multiplicities greater than one. It is well known that a closed form solution exists for computing coefficients of RGFs. Also, one can write the linear recurrence relation associated with every RGF into a matrix format. Using matrices, one can compute coefficients for an RGF, such as Molien series for finite groups, in logarithmic time. What has not yet been shown (or is not yet commonly discussed) is that one can conceptualize an RGF as a system of connected cycles within an overarching adjacency matrix. For example, a single cycle of length two would have vertex A connect to vertex B which itself connects back to vertex A with a directed arrow of weight $c_i$. In this conceptualization, each coefficient of an RGF can be reproduced by taking a suitable adjacency matrix to an integer power. Nothing essential is lost by taking this perspective. Due to the self-similar nature of the matrix, we devise an algorithm that can calculate coefficients of RGFs in constant time. Using memoization, a technique for caching intermediate results, calculating coefficients of RGFs can also be done in logarithmic time. One observation is that, depending on the situation (i.e. what $q(x)$ is), there may be a computational benefit to taking the cyclical perspective. For example, for certain $q(x)$, the traditional matrix has cells containing positive and negative values whereas the cyclical approach has cells containing only positive values. The computational benefit is probably irrelevant for computers; however, it may be important for restrictive systems, such as biological systems / neural networks that may have a tight operating envelope. We make a final observation that each cyclical matrix representation can be thought of as a graph which is an epsilon away from being strongly connected. Studying the behavior of these matrices may yield insight into the behavior of a broader class of function. In essence, the study of sequences modeled by RGFs can be converted to the study of connected cyclical graphs that model the RGFs or vice versa.

A partial matrix A is a rectangular array with entries in F ∪ {∗}, where F is the ground field, and ∗ is a placeholder symbol for entries which are not specified. The minimum rank mr(A) is the smallest value of the ranks of all matrices obtained from A by replacing the ∗ symbols with arbitrary elements in F. For any bipartite graph G with vertices (U, V), one defines the set M(G) of partial matrices in which the row indexes are in U, the column indexes are in V, and the (u, v) entry is specified if and only if u, v are adjacent in G. We prove that, if G is chordal bipartite, then the minimum rank of any matrix in M(G) is determined by the ranks of its fully specified submatrices. This result was conjectured by Cohen, Johnson, Rodman, Woerdeman in 1989.

In this paper we introduce and develop the notion of universe, induced communities and cells with their corresponding spots. We study the concept of the density, the mass of communities, the concentration of spots in a typical cell, connectedness and the rotation of communities. In any case we establish the connection that exist among these notions. We also formulate the celebrated union-close set conjecture in the language of density of spots and the mass of a typical community.

In this paper we introduce the notion of universe, induced communities and cells with their corresponding spots. Using this language we formulate and prove the union close set conjecture by showing that for any finite universe $\mathbb{U}$ and any induced community $\mathcal{M}_{\mathbb{U}}$ there exist some spot $a\in \mathbb{U}$ such that the density \begin{align} \mathcal{D}_{\mathcal{M}_{\mathbb{U}}}(a)\geq \frac{1}{2}.\nonumber \end{align}

This paper deals with nite (possibly not complete) unambiguous automata, not necessarily deterministic. In this setting, we investigate the problem of the minimal length of the uncompletable word. This problem is associated with the well-known conjecture formulated by A. Restivo. We introduce the concept of relatively maximal row for a suitable set of matrices, and show the existence of a relatively maximal row of length of quadratic order with respect to the number of the states of the treated automaton. We give some estimates of the maximal length of the minimal uncompletable word in connection with the number the states of the involved automaton and the length of a suitable relatively maximal but not maximal word, provided that it exists. In the general case, we establish an estimate of the length of the minimal uncompletable word in terms of the number of states of the studied automaton, the length of a suitable relatively maximal word and the minimal length of the uncompletable word of the automaton formed by all associated maximal rows.

Let $G$ be a graph containing no component isomorphic to the path graph of order $2$. Denote by $d_w$ the degree of a vertex $w$ in $G$. The augmented Zagreb index ($AZI$) of $G$ is the sum of the quantities $(d_ud_v/(d_u+d_v-2))^3$ over all edges $uv$ of $G$. Denote by $\mathcal{G}(n,\chi)$ the class of all connected graphs of a fixed order $n$ and with a fixed chromatic number $\chi$, where $n\ge5$ and $3\le \chi \le n-1$. The problem of finding graph(s) attaining the maximum $AZI$ in the class $\mathcal{G}(n,\chi)$ has been solved recently in [F. Li, Q. Ye, H. Broersma, R. Ye, MATCH Commun. Math. Comput. Chem. 85 (2021) 257--274] for the case when $n$ is a multiple of $\chi$. The present paper gives the solution of the aforementioned problem not only for the remaining case (that is, when $n$ is not a multiple of $\chi$) but also for the case considered in the aforesaid paper.

In this paper, we have derived a formula to find combinatorial sums of the type $\sum_{r=0}^n r^k {n\choose r}$ for $k \in \mathbb{N}$. The formula is conveniently expressed as a linear combination of terms involving the falling factorial. The co-efficients in this linear expression satisfy a recurrence relation, which is identical to that of the Stirling numbers of the first and second kind.

A Motzkin Path is a walk left-to-right starting at the horizontal axis, consisting of up, down or horizontal steps, never descending below the horizontal axis, and finishing at the horizontal axis. Interpret Motzkin Paths as vertical geologic cuts through mountain ranges with limited slopes. The natural embedding of these paths defines Motzkin Islands as sets of graphs labeled on vertices by non-negative integers (altitudes), a graph cycle defining a shoreline at zero altitude, and altitude differences along edges never larger than one. We address some of these islands with simple shapes on triangular and quadratic meshes.

In this short note, we give a proof for the fact that the chromatic number of the EFL graph formed by the adjoining of k cliques such that any two cliques share at most one vertex is k

In this article, by algebraic and geometrical techniques, I give a proof to the famous Ulam's conjecture on the (-1)-reconstruction of the symmetric graphs with at least 3 elements conjectured in 1942, although was published only in 1960.

I study the link between the adjoint action and the Hamiltonian cycles in a symmetric graph. Then by a simple algebraic resolution of a system of equations with several variables I find all the Hamiltonian cycles of the graph. Finally I will apply the results found to give an algorithm of order $ \mathcal{ O } (n ^ 3 ) $ allowing to quickly give all the Hamiltonian cycles with their distance. This gives a proof of the conjecture $ P = NP $.

We introduce and study the needle \begin{align}(\Gamma_{\vec{a}_1} \circ \mathbb{V}_m)\circ \cdots \circ (\Gamma_{\vec{a}_{\frac{l}{2}}}\circ \mathbb{V}_m):\mathbb{R}^n\longrightarrow \mathbb{R}^n.\nonumber \end{align} By exploiting the geometry of compression, we prove that this function is a function modeling an $l$-step self avoiding walk for $l\in \mathbb{N}$. We show that the total length of the $l$-step self-avoiding walk modeled by this function is of the order \begin{align}\ll \frac{l}{2}\sqrt{n}\bigg(\mathrm{\max}\{\mathrm{sup}(x_{j_k})\}_{\substack{1\leq j\leq \frac{l}{2}\\1\leq k\leq n}}+\mathrm{\max}\{\mathrm{sup}(a_{j_k})\}_{\substack{1\leq j\leq \frac{l}{2}\\1\leq k\leq n}}\bigg)\nonumber \end{align}and at least \begin{align}\gg \frac{l}{2}\sqrt{n}\bigg(\mathrm{\min}\{\mathrm{Inf}(x_{j_k})\}_{\substack{1\leq j\leq \frac{l}{2}\\1\leq k\leq n}}+\mathrm{\min}\{\mathrm{Inf}(a_{j_k})\}_{\substack{1\leq j\leq \frac{l}{2}\\1\leq k\leq n}}\bigg).\nonumber \end{align}

In this paper we study the distribution of boundary points of expansion. As an application, we say something about the lonely runner problem. We show that given $k$ runners $\mathcal{S}_i$ round a unit circular track with the condition that at some time $||\mathcal{S}_i-\mathcal{S}_{i+1}||=||\mathcal{S}_{i+1}-\mathcal{S}_{i+2}||$ for all $i=1,2\ldots,k-2$, then at that time we have \begin{align}||\mathcal{S}_{i+1}-\mathcal{S}_i||>\frac{\mathcal{D}(n)\pi}{k-1}\nonumber \end{align}for all $i=1,\ldots, k-1$ and where $\mathcal{D}(n)>0$ is a constant depending on the degree of a certain polynomial of degree $n$. In particular, we show that given at most eight $\mathcal{S}_i$~($i=1,2,\ldots, 8$) runners running round a unit circular track with distinct constant speed and the additional condition $||\mathcal{S}_i-\mathcal{S}_{i+1}||=||\mathcal{S}_{i+1}-\mathcal{S}_{i+2}||$ for all $1\leq i\leq 6$ at some time $s>1$, then at that time their mutual distance must satisfy the lower bound\begin{align}||\mathcal{S}_{i}-\mathcal{S}_{i+1}||>\frac{\pi}{7C\sqrt{3}}\nonumber \end{align}for some constant $C>0$ for all $1\leq i \leq 7$.

In this manuscript, the operations on neutrosophic vague graphs are introduced. Moreover, Cartesian product, cross product, lexicographic product, strong product and composition of neutrosophic vague graph are investigated and the proposed concepts are illustrated with examples.

We introduce and study the needle function. We prove that this function is a function modeling $n$-step self avoiding walk. We show that the total length of the $l$-step self-avoiding walk modeled by this function is of the order \begin{align}\ll \frac{n^{\frac{3}{2}}}{2}\bigg(\mathrm{\max}\{\mathrm{sup}(x_j)\}_{1\leq j\leq \frac{l}{2}}+\mathrm{max}\{\mathrm{sup}(a_j)\}_{1\leq j\leq \frac{l}{2}}\bigg).\nonumber \end{align}

In this lecture we count the number of integer partitions P(n) using an elementary algorithm based on the combinatorics of trees, coded using free apps on the iPad.

Given two sets, one consisting of variables representing distinct positive n numbers, the other set `a kind of power set' of this n-element set. I got interested in the fact that for the latter set, depending on the values of two elements, it can occur that not every pair of elements is `comparable', that is to say, it is not always uniquely determined which of two elements is smaller. By proving theorems in order to go ahead with our research, we show a table which describes for how many `comparable' cases exist, for several n's.

An Hadamard matrix is a square matrix $Hin M_N(pm1)$ whose rows and pairwise orthogonal. More generally, we can talk about the complex Hadamard matrices, which are the square matrices $Hin M_N(mathbb C)$ whose entries are on the unit circle, $|H_{ij}|=1$, and whose rows and pairwise orthogonal. The main examples are the Fourier matrices, $F_N=(w^{ij})$ with $w=e^{2pi i/N}$, and at the level of the general theory, the complex Hadamard matrices can be thought of as being some sort of exotic, generalized Fourier matrices. We discuss here the basic theory of the Hadamard matrices, real and complex, with emphasis on the complex matrices, and their geometric and analytic aspects.

The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, is called 'the grasshopper problem'. To this problem Kos developed theory from unique viewpoints by reference of Noga Alon’s combinatorial Nullstellensatz. We have tried to solve this problem by an original method inspired by a polynomial function that Kos defined, then examined for n=3, 4 and 5. For almost cases the claim of this problem follows, but there remains imperfection due to 'singularity'.

This work provides a Java program which constructs free polyominoes of size n sorted by width and height of the convex hull (i.e., its rectangular bounding box). The results correct counts for 15-ominoes published in the 1967 proceedings of the SIAM Fall Meeting, and extend them to 17-ominoes and partially to even larger polyominoes. [vixra:1905.0474]

In this study we answer questions that have to do with finding out the total number of ways of arranging a finite set of symbols or objects directly under a single line constraint set of finite symbols such that common symbols between the two sets do not repeat on the vertical positions. We go further to study the outcomes when both sets consist of the same symbols and when they consist of different symbols. We identify this form of permutation as 'second-order permutation' and show that it has a corresponding unique factorial which plays a prominent role in most of the results obtained. This has been discovered by examining many practical problems which give rise to the emergence of second-order permutation. Upon examination of these problems, it becomes clear that a directly higher order of permutation exist. Hence this study aims at equipping mathematics scholars, educators and researchers with the necessary background knowledge and framework for incorporating second-order permutation into the field of combinatorial mathematics.

Previously, a determination of the relationship between the Natural numbers (N) and the n'th odd fibbinary number has been made using a relationship with the Golden ratio \phi=(Sqrt[5]+1}/2 and \tau=1/\phi. Specifically, if the n'th odd fibbinary equates to the j'th N, then j=Floor[n(\phi+1) - 1]. This note documents the completion of the relationship for the even fibbinary numbers, such that if the n'th even fibbinary equates to the j'th N, then j=Floor[n(\tau+1) + \tau].

The study of perebor dates back to the Soviet-era mathematics, especially in the 1980s [1]. Post-Soviet mathematicians have been working on many problems in combinatorial optimization. One of them is Maximum Edge Biclique Problem (MBP). In [2], the author proves that MBP is NP-complete. In this note, we give a polynomial time algorithm for MBP by using linear programming (LP). Thus, some flaw needs to be found in Peeter's work. We leave this to the community.

The study of perebor dates back to the Soviet-era mathematics, especially in the 1980s [1]. Post-Soviet mathematicians has been working on many problems in combinatorial optimization. One of them is Exact Weight Perfect Matching of Bipartite Graph (EWPM).This particular problem has been thoroughly considered by [2], [3], [4]. In this note, we give a simpler proof about the solvability of EWPM.

This proves that any even number larger than 2 can be written as the sum of two prime Numbers, also known as the "goldbach conjecture" or "goldbach conjecture about the even" is in the test for any greater than or equal to 6 even conform to guess the number of prime Numbers, accidentally discovered the prime Numbers of "additionality" and further expansion of verification.This article does not focus on the functional expressions of prime Numbers themselves, but takes a different approach to prove that all even Numbers can be composed of two prime Numbers

In this paper, we present several models for Capacitated Vehicle Routing Problem (CVRP) using Choco solver. A concise introduction to the constraint programming methods is included. Then, we construct two models for CVRP. Experimental results for each model are given in details.

The non-cyclic graphs known as trees may be labeled by assigning positive integer numbers (weights) to their vertices or to their edges. We count the trees up to 10 vertices that have prescribed sums of weights, or, from the number-theoretic point of view, we count the compositions of positive integers that are constrained by the symmetries of trees.

This paper is concerned with the problem of exact MAP inference in general higher-order graphical models by means of a traditional linear programming relaxation approach. In fact, the proof that we have developed in this paper is a rather simple algebraic proof being made straightforward, above all, by the introduction of two novel algebraic tools. Indeed, on the one hand, we introduce the notion of delta-distribution which merely stands for the difference of two arbitrary probability distributions, and which mainly serves to alleviate the sign constraint inherent to a traditional probability distribution. On the other hand, we develop an approximation framework of general discrete functions by means of an orthogonal projection expressing in terms of linear combinations of function margins with respect to a given collection of point subsets, though, we rather exploit the latter approach for the purpose of modeling locally consistent sets of discrete functions from a global perspective. After that, as a first step, we develop from scratch the expectation optimization framework which is nothing else than a reformulation, on stochastic grounds, of the convex-hull approach, as a second step, we develop the traditional LP relaxation of such an expectation optimization approach, and we show that it enables to solve the MAP inference problem in graphical models under rather general assumptions. Last but not least, we describe an algorithm which allows to compute an exact MAP solution from a perhaps fractional optimal (probability) solution of the proposed LP relaxation.

We create the unlabeled or vertex-labeled graphs with up to 10 edges and up to 10 vertices and classify them by a set of standard properties: directed or not, vertex-labeled or not, connectivity, presence of isolated vertices, presence of multiedges and presence of loops. We present tables of how many graphs exist in these categories.

Let G1 and G2 be two graph with vertex sets V (G1); V (G2) and edge sets E(G1);E(G2) respectively. The subdivision graph S(G) of a graph G is the graph obtained by inserting a new vertex into every edges of G. The SGvertexjoin of G1 and G2 is denoted by G1}G2 and is the graph obtained from S(G1) [ G1 and G2 by joining every vertex of V (G1) to every vertex of V (G2). In this paper we determine the adjacency spectra ( respectively Laplacian spectra and signless Laplacian spectra) of G1}G2 for a regular graph G1 and an arbitrary graph G2

The Duplication graph DG of a graph G, is obtained by inserting new vertices corresponding to each vertex of G and making the vertex adja- cent to the neighbourhood of the corresponding vertex of G and deleting the edges of G. Let G1 and G2 be two graph with vertex sets V (G1) and V (G2) respectively. The DG - vertex join of G1 and G2 is denoted by G1 t G2 and it is the graph obtained from DG1 and G2 by joining every vertex of V (G1) to every vertex of V (G2). The DG - add vertex join of G1 and G2 is denoted by G1 ./ G2 and is the graph obtained from DG1 and G2 by joining every additional vertex of DG1 to every vertex of V (G2). In this paper we determine the A - spectra and L - spectra of the two new joins of graphs for a regular graph G1 and an arbitrary graph G2 . As an application we give the number of spanning tree, the Kirchhoff index and Laplace energy like invariant of the new join. Also we obtain some infinite family of new class of integral graphs

We present a spectral theory of uniform, regular and linear hyper- graph. The main result are the nature of the eigen values of (k; r) - regular linear hypergraph and the relation between its dual and line graph. We also discuss some properties of Laplacian spectrum of a (k; r) - regular hypergraphs.

This is a numerical study of the combinatorial problem of packing hexagons of some equal size into a larger hexagon. The problem is well defined if all hexagon edges have integer length and if their centers and vertices share the common lattice points of a triangular grid with unit distances.

A terrace form assigns an integer altitude to each point of a finite two-dimensional square grid such that the maximum altitude difference between a point and its four neighbors is one. It is 2-way monotonic if the sign of this altitude difference is zero or one for steps to the East or steps to the South. We provide tables for the number of 2-way monotonic terrace forms as a function of grid size and maximum altitude difference, and point at the equivalence to the number of 3-colorings of the grid.

The multinomial coefficients count the number of ways (of permutations) of placing a number of partially distinguishable objects on a line, taking ordering into account. A well-known two-parametric family of counts arises if there are objects of c distinguishable colors and m objects of each color, m*c objects in total, to be placed on line. In this work we propose an algorithm to count the permutations where no two objects of the same color appear side-by-side on the line. This eliminates all permutations with "clusters" of colors. Essentially we represent filling the line sequentially with objects as a tree of states where each node matches one partially filled line. Subtrees are merged if they have the same branching structure, and weights are assigned to nodes in the tree keeping track of how many mergers take place. This is implemented in a JAVA program; numerical results confirm Hardin's earlier counts for this kind of restricted permutations.

We provide a C++ program which searches for the smallest number of pouring steps that convert a set of jugs with fixed (integer) capacities and some initial known (integer) water contents into another state with some other prescribed water contents. Each step requires to pour one jug into another without spilling until either the source jug is empty or the drain jug is full-because the model assumes the jugs have irregular shape and no further marks. The program simply places the initial jug configuration at the root of the tree of state diagrams and deploys the branches (avoiding loops) recursively by generating all possible states from known states in one pouring step.

Erdős-Szekeres Theorem is proven. The proof is very similar to the original given by Erdős and Szekeres. However, it explicitly uses properties of binary trees to prove and visualize the existence of a monotonic subsequence. It is hoped that this presentation is helpful for pedagogical purposes.

This is a course on advances in graph algorithms that we taught in Taiwan. The topics included are Exact algorithms, Graph classes, fixed-parameter algorithms, and graph decompositions.

This paper shows the non-existence of magic square of squares in order three by investigating two new tools, the first is representing three perfect squares in arithmetic progression by two numbers and the second is realizing the impossibility of two similar equations for the same problem at the same time in different ways and the variables of one is relatively less than the other.

A self-avoiding walk (SAW) is a path on a lattice that does not pass through the same point more than once. We develop a method for enumerating self-avoiding walks in a lattice by decomposing them into smaller pieces called tiles, solving particular cases on the square, triangular and cubic lattices. We also show that enumeration of SAWs in a lattice is related to enumeration of edge-connected shapes, for example polyominoes.

Let G and H be two cographs. We show that the problem to determine whether H is a retract of G is NP-complete. We show that this problem is fixed-parameter tractable when parameterized by the size of H. When restricted to the class of threshold graphs or to the class of trivially perfect graphs, the problem becomes tractable in polynomial time. The problem is also solvable in linear time when one cograph is given as an induced subgraph of the other. We characterize absolute retracts for the class of cographs. Foldings generalize retractions. We show that the problem to fold a trivially perfect graph onto a largest possible clique is NP-complete. For a threshold graph this folding number equals its chromatic number and achromatic number.

Let GG be a class of graphs. A graph G is a probe graph of GG if its vertex set can be partitioned into a set P of `probes' and an independent set N of `nonprobes' such that G can be embedded into a graph of GG by adding edges between certain nonprobes. In this book we investigate probe graphs of various classes of graphs.

We show that there is a linear-time algorithm to partition the edges of a planar graph into triangles. We show that the problem is also polynomial for toroidal graphs but NP-complete for k-planar graphs, where k is at least 8.

In memoriam N.G. de Bruijn. In this article I present some highlights of De Bruijn's contributions in combinatorics. This article does not survey his work on eg Penrose tilings, asymptotics or AUTOMATH; other surveys on these topics are being written by others.

We build unbounded classes of plane and projective plane multiwheels that are 4-critical that are received summing odd wheels as edge sums modulo two. These classes can be considered as ascending from single common graph that can be received as edge sum modulo two of the octahedron graph O and the minimal wheel W3. All graphs of these classes belong to 2n - 2-edges-class of graphs, among which are those that quadrangulate projective plane, i.e., graphs from Groetzsch class, received applying Mycielski's Construction to odd cycle.

We build unbounded classes of plane and projective plane multiwheels that are 4-critical that are received summing odd wheels as edge sums modulo two. These classes can be considered as ascending from single common graph that can be received as edge sum modulo two of the octahedron graph O and the minimal wheel W_3.

fix a ship�s position in charted interstellar space with the assistance of a three dimensional computer based stellar chart and star camera spectrometers capable of measuring angular separations between three sets of pair stars. The method offers another tool for the navigator to rely on if alternative position fixing methods are not available or if the navigator wishes to verify the validity of one�s position given by other means.

We consider combinatorial maps with fixed combinatorial knot numbered with augmenting numeration called normalized knot. We show that knot's normalization doesn't affect combinatorial map what concerns its generality. Knot's normalization leads to more concise numeration of corners in maps, e.g., odd or even corners allow easy to follow distinguished cycles in map caused by the fixation of the knot. Knot's normalization may be applied to edge structuring knot too. If both are normalized then one is fully and other partially normalized mutually.