Magic Sum Research Articles

Constant Sum Group Flows of Graphs

As an analogous concept of a nowhere-zero flow for directed graphs, zero-sum flows and constant-sum flows are defined and studied in the literature. For an undirected graph, a zero-sum flow (constant-sum flow resp.) is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero (constant h resp.), and we call it a zero-sum k-flow (h-sum k-flow resp.) if the values of the edges are less than k. We extend these concepts to general constant-sum A-flow, where A is an Abelian group, and consider the case A=Zk the additive Abelian cyclic group of integer congruences modulo k with identity 0. In the literature, a graph is alternatively called Zk-magic if it admits a constant-sum Zk-flow, where the constant sum is called a magic sum or an index for short. We define the set of all possible magic sums such that G admits a constant-sum Zk-flow to be Ik(G) and call it the magic sum spectrum, or for short, the index set of G with respect to Zk. In this article, we study the general properties of the magic sum spectrum of graphs. We determine the magic sum spectrum of complete bipartite graphs Km,n for m≥n≥2 as the additive cyclic subgroups of Zk generated by kd, where d=gcd(m−n,k). Also, we show that every regular graph G with a perfect matching has a full magic sum spectrum, namely, Ik(G)=Zk for all k≥3. We characterize a 3-regular graph so that it admits a perfect matching if and only if it has a full magic sum spectrum, while an example is given for a 3-regular graph without a perfect matching which has no full magic sum spectrum. Another example is given for a 5-regular graph without a perfect matching, which, however, has a full magic sum spectrum. In particular, we completely determine the magic sum spectra for all regular graphs of even degree. As a byproduct, we verify a conjecture raised by Akbari et al., which claims that every connected 4k-regular graph of even order admits a 1-sum 4-flow. More open problems are included.

Магический квадрат Лошу: регистр истины

The world’s oldest 3x3 magic square, discovered/invented in Ancient China and now known as Luoshu 洛書/ ‘Document [from the River] Luo’, was endowed by Chinese tradition with unprecedented dignity and placed at the very heart of Chinese thought. The bewitching geometric-numerical imagery of Luoshu, open to a great many different visions, when reading-interpretation becomes the final moment of the very act of perception, turns this Chinese mandala into a real eye trap. With its disturbing persistence, it resembles the magically attractive ‘Zahir’ from Borges’ short story of the same name. Among the most diverse ritual and ideological instrumentalizations of the Luoshu magic square (from the sacred emblem of cosmic harmony to the requisite of a geomancer), the mobilization of this esoteric figure for the arithmetization of the cornerstone of all Chinese philosophy, the fundamentally non-verbalizable Tao, is dominant. The coding of the dao by the number 15 is reinforced by its spatialization, so that the entire “Document [from the river] Lo” appears as a map of the various trajectories of the Tao within this nine-field square. Moreover, the coincidence (in number 15) of differently composed sums appears as an inscrutable variety of paths leading to the same goal - the final implementation of Tao. By virtue of the validity of the equality 15 =mod10 5 the magic sum of the Luoshu square (Const15) immutably, although covertly (in the form of a number 5), centers the entire Luoshu configuration. This secrecy of the magical constant (the latter is absent in the entire observable space of the “Document [from the river] Lo”) refers to the hidden “back-side” of Luoshu numbers, represented by the number 10 (in its role as a modulus of comparison in the arithmetic of residues modulo 10). Judging by the directly visible, so to speak, ‘front’ part of the magic square of order 3, it is allowed to count only up to nine in it. But the already absent-present magic sum (number 15), breaking the seemingly inescapable circle of arithmetic of residues, brings to light the comparison module (number 10) as Luosh’s hidden ‘truth’, which alone gives meaning to the entire nine-cell construction. Awareness of this truth is the first step in the transition to the ‘register of truth’ of this extraordinary gestalt. The subsequent connection to it of problems focused by the Pythagorean theorem radically expands Luoshu’s ‘register of truth’. The geometrized arithmetic of the Luoshu magic square, which is a unique spatial-numerical fixation of the seemingly fundamentally non-objectivable Tao (i.e., combining the apparently incompatible) marks the possibility of a paradoxical union of Heraclitus with Pythagoras, successfully realized by the Chinese tradition.

On magic distinct labellings of simple graphs

A magic labelling of a graph G with magic sum s is a labelling of the edges of G by nonnegative integers such that for each vertex v∈V, the sum of labels of all edges incident to v is equal to the same number s. Stanley gave remarkable results on magic labellings, but the distinct labelling case is much more complicated. We consider the complete construction of all magic labellings of a given graph G. The idea is illustrated in detail by dealing with three regular graphs and a non-regular graph. We give combinatorial proofs. The structure result was used to enumerate the corresponding magic distinct labellings. The idea can be carried out directly for graphs with no more than 9 edges.

A GENERALIZATION OF NONSINGULAR REGULAR MAGIC SQUARES

A generalization of regular magic squares with magic sum $\mu$ is an sq-corner (or square corner) magic square. It is a magic square satisfying the condition that the sum of 4 entries, square symmetrically placed with respect to the center, equals $\frac{4\mu}{n}$. Using the sq-corner magic squares of order $n$, a construction of sq-corner magic squares of order {$n+2$} is derived. Moreover, this construction provides some nonsingular classical sq-corner magic squares of all orders. In particular, a nonsingular regular magic square of any odd order can be constructed under this new method, as well.

AN ESTIMATOR OF MOMENT OF INERTIA OF MAGIC SQUARE ARRAY OF MASSES

The classical definition of magic square of order is a square array of consecutive natural numbers from 1 to such that the raw, column and diagonal sum add up to the same number. When the magic square entries are considered as an array of masses of a rigid body, it is established that its moment of inertia is a function of This work consider a more general magic array of masses of a rigid body to establish its inertial moment as a function of the magic sum and the central entry of the array. The paper further discusses the advantage of this development.

Research on Magic Square Construction Based on Genetic Algorithm

The construction of magic squares is a complex permutation and combination problem. It is very difficult to construct the magic square with the continuous minimum natural number, and its solution has the nature of diversity and increases exponentially with the increase of order. In this paper, the magic square is constructed by selecting specific real numbers in different situations. Firstly, the magic square model is realized by using genetic algorithm, and the population is selected, crossed and mutated, and the optimal solution is selected, so that the semi-magic square whose sum of rows and columns is magic sum can be generated. Then, the parent gene of diagonal magic population is generated again by row exchange and column exchange, and the data of diagonal magic population is improved. Then, the selection, crossover and diagonal local adjustment are carried out in turn, and finally, the N-order high-quality magic square in the interval of [1, N²] natural number is obtained.

A block cipher algorithm developed using Magic Square in the seventh Order

This paper use the magic square of order 7 to develop Block encryption algorithms that work for encoding color images and text in GF(P) and GF(28). The algorithm use key with length = 35 and a message with length =14 are used, First, the keys are placed in the previously agreed positions and the remaining positions remain for the message, then the magic sum for each of them is calculated. Complexity, velocity, NIST calculations, and histogram calculations were calculated and the results were compared with the 5th degree magic square, where the complexity was with using GF(P) = (256)14 × (P)35 and using GF(28) = (256)14 × (256)35, The magic square of the seventh degree is better than the magic square of the fifth degree in cryptography in terms of complexity and a slight difference in speed.

A New Block Cipher Algorithm Using Magic Square of Order Five and Galois Field Arithmetic with Dynamic Size Block

<p class="0abstract">This paper describes the development of encryption algorithms using the magic square of order 5 and Multi-level keys with the addition of Matrix keys to increase implementation speed and complexity. This work relied mainly on the magic sum and some equations that were added as an improvement on previous work. Multi-level keys were used for three different message sizes, and an additional key matrix with size 5×5 was used to add more complexity. The proposed work was performed using both GF(P) and GF(2<sup>8</sup>). Results were compared with the MS3, they have been found good, with acceptable speed and high complexity where it was (P)<sup>9</sup> × (256)<sup>16</sup> in the first algorithm, (P)<sup>9</sup> × (256)<sup>16</sup> × 3 in the second algorithm, and (P)<sup>9</sup> × (256)<sup>16</sup> × 3 × (P)<sup>25</sup> in the third algorithm, the complexity changed according to the chosen value of N randomness, in addition to speed, complexity, NIST calculations have been performed for texts and histogram calculations for different images were calculated and compared as well.</p>

Magic-Matrix-Based Power Allocation for Downlink Nonorthogonal Multiple Access

In a wireless cellular system, resource allocation among active users is a challenging task. The cellular systems with nonorthogonal multiple access (NOMA) solve the problem of bandwidth and time allocation. All the active users in the same NOMA cluster are allowed to access the complete cluster bandwidth at the same time. However, power allocation in NOMA is a multiconstraint optimization problem. The complexity of power allocation increases with the increase in the number of users inside the cluster. This article presents a novel technique for power allocation in a NOMA downlink cluster, using the magic sum of the magic matrix. The sum throughput (SUMTP) and the min–max fairness index (MMFI) are calculated and compared with an existing method for the downlink NOMA system. The proposed method shows comparable SUMTP performance to the conventional optimized method while ensuring better MMFI. We observed an inverse association between SUMTP and MMFI. In addition, the execution time of the proposed method is much lesser than that of the conventional optimized method owing to lower number of flops required for magic-matrix-based power allocation.

Unified Construction of Normal Bimagic Squares of Doubly Even Orders Based on Quasi Bimagic Pairs

A general magic square of order n is an n × n matrix consisting of integers in such a way that the sum of all elements in each row, each column, main diagonal and back diagonal is the same number called the magic sum of this matrix. A general magic square of order 2n is normal if its entries are 4n 2 consecutive odd integers 1−4n 2, 3−4n 2,…,4n 2 −3, 4n 2 −1. A normal bimagic square of order 4n is a normal magic square such that the sum of squares of all elements in each row, each column, main diagonal and back diagonal is the same number. Using the reflection matrix R and a quasi bimagic pair (A,B) where A and B are two special 2n × 2n matrices consisting of odd integers, we give a unified and very simple construction of normal bimagic square H of order 4n for all n ≥ 2 : . We construct a quasi bimagic pair by means of orthogonal diagonal latin squares for n ≠ 3 and by means of the computer seeking for n = 3.

On H-supermagic labeling of friendship edge corona product with path and sun edge corona product with path

Let G be a simple, finite, connected and undirected graph with vertex set V and edge set E. Graph G admits an H-covering if every edge in E belongs to a subgraph of G isomorphic to H. Graph G is H− magic if there is a total labeling , such that each subgraph H′ = (V′, E′) of G isomorphic to H and satisfying where m(f) is a constant magic sum. Additionaly, G admits H− supermagic if f(V ) = {1, 2, …, |V(G)|}. The edge corona product between graph G1 and G2 is a graph obtained by taking one copy of G1 and |E(G1)| copies of G2 and then joining two end − vertices of the ith edge of G1 to every vertex in ith copy of G2. This research provides C3 ◊ Pn-supermagic labeling on fn ◊ Pn and P3 ◊ Pn-supermagic labeling on Sn ◊ Pn.

H− Supermagic labeling on domino ʘ Pn and Cn ◊ Sn

A finite simple graph G with vertex set V and edge set E admits an H-covering if every edge in E belongs to a subgraph of G isomorphic to H. We said G to be an H magic if there exists a total labeling , such that for each subgraph H’ = (V’, E’) of G isomorphic to H, ∑v∈V f(v) + ∑e∈E f(e) = m(f) where m(f) is a constant magic sum. Then G is H supermagic if f(V ) = {1, 2, …, |V (G)|}. The edge corona product between graph G1 and G2 is a graph obtained by taking one copy of G1 and |E(G1)| copies of G2 and then joining two end − vertices of the ith edge of G1 to every vertex in ith copy of G2. This research focuses on C4 ʘ Pn− supermagic labeling on Domino ʘ Pn and P2 ◊ Sn supermagic labeling on Cn ◊ Sn.

Magic Polygons and Degenerated Magic Polygons: Characterization and Properties

In this work we define Magic Polygons P(n, k) and Degenerated Magic Polygons D(n, k) and we obtain their main properties, such as the magic sum and the value corresponding to the root vertex. The existence of magic polygons P(n, k) and degenerated magic polygons D(n, k) are discussed for certain values of n and k.

On P2 ⋄ Pn-supermagic labeling of edge corona product of cycle and path graph

A simple graph G = (V, E) admits a H-covering, where H is subgraph of G, if every edge in E belongs to a subgraph of G isomorphic to H. Graph G is H-magic if there is a total labeling , such that each subgraph of G isomorphic to H and satisfying where m(f) is a constant magic sum. Additionaly, G admits H-supermagic if . The edge corona of Cn and Pn is defined as the graph obtained by taking one copy of Cn and n copies of Pn, and then joining two end-vertices of the i-th edge of Cn to every vertex in the i-th copy of Pn. This research aim is to find H-supermagic covering on an edge corona product of cycle and path graph where H is . We use k-balanced multiset to solve our reserarch. Here, we find that an edge corona product of cycle and path graph is supermagic for n > 3.

H-Supermagic Labeling on Coronation of Some Classes of Graphs with a Path

A simple graph G = (V; E) admits an H-covering if every edge in E belongs to a subgraph on G that isomorphic to H. A graph G is H-magic if there is exists a total labeling f : V (G)∪E(G) → {1, 2, …, |V|+|E|}, such that each subgraph H′ = (V′(H′); E′(H′)) on G satisfies f(H′) = ∑v∈V′ f(v) + ∑e∈E′ f(e) = m(f), where m(f) is a constant magic sum. A graph G is a H-supermagic labeling if f(V ) = {1, …, |V|} and s(f) is a constant supermagic sum. The research aims to study a H-supermagic labeling on corona product of a star graph with a path and a wheel graph with a path. We prove that the corona product of a star graph with a path Sn ⊙ Pm is a Um,2-supermagic for m is odd or m, n are even and m ≥ 3 and the corona product of a wheel graph with a path Wn ⊙ Pm is a C3 ⊙ Pm-supermagic for m ≥ 3.

Public Key Cipher with Signature Based on Diffie-Hellman and the Magic Square Problem

In the present paper, wedeveloped a new variant of asymmetric cipher (Public Key) algorithm based on the Discrete Logarithm Problem (DLP) uses Diffie-Hellman key exchange protocol and the mathematical features of magic square. The proposed method exploits the DLP problem in the key exchange by using the final value of Diffie-Hellman key agreement, just as the dimension to the magic construction and through which determines the type of magic square construction if it is (odd, singly-even or doubly-even) magic square as well as through which determines starting number and the difference value. From the other point, it exploits the magic squares problem in encryption/decryption and signing/verifyingoperations. The developed methodextremely speed in the Encryption/Decryption process as well as in the digital signaturethat uses aSecure Hash Algorithm SHA-1, since the proposed cipher does not use logarithm and the factorization as the traditional algorithms in ciphering and deciphering operations just in the mutual exchanging, but it depends mainly upon the magic constant and magic sum as alternate that deduced from the result of multiplied fixed value, which selected randomly and imposesto keep secret, as we shall explained in the next sections.

Sum rules via large deviations

In the theory of orthogonal polynomials, sum rules are remarkable relationships between a functional defined on a subset of all probability measures involving the reverse Kullback–Leibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction. Killip and Simon [24] have given a revival interest to this subject by showing a quite surprising sum rule for measures dominating the semicircular distribution on [ − 2 , 2 ] . This sum rule includes a contribution of the atomic part of the measure away from [ − 2 , 2 ] . In this paper, we recover this sum rule by using probabilistic tools on random matrices. Furthermore, we obtain new (up to our knowledge) magic sum rules for the reverse Kullback–Leibler divergence with respect to the Marchenko–Pastur or Kesten–McKay distributions. As in the semicircular case, these formulas include a contribution of the atomic part appearing away from the support of the reference measure.

A Problem on Edge-magic Labelings of Cycles

AbstractIn 1970, Kotzig and Rosa defined the concept of edge-magic labelings as follows. Let G be a simple (p; q)-graph (that is, a graph of order p and size q without loops or multiple edges). A bijective function f : V(G)⊔E(G) → {1; 2, … p+q} is an edge-magic labeling of G if f (u)+ f (uv)+ f (v) = k, for all uv ∊ E(G). A graph that admits an edge-magic labeling is called an edge-magic graph, and k is called the magic sum of the labeling. An old conjecture of Godbold and Slater states that all possible theoretical magic sums are attained for each cycle of order n ≥ 7. Motivated by this conjecture, we prove that for all n0∊ N, there exists n 2 N such that the cycle Cn admits at least n0 edge-magic labelings with at least n0 mutually distinct magic sums. We do this by providing a lower bound for the number of magic sums of the cycle Cn, depending on the sum of the exponents of the odd primes appearing in the prime factorization of n.

합의 범위를 이용한 지수귀문도 해의 탐구

Jisuguimundo is an inimitable magic hexagon devised by Choi Seok-Jeong, who was the author of GuSuRyak as well as a prime minister in Joseon dynasty. Jisuguimundo, recorded in GuSuRyak, is also known as Hexagonal Tortoise Problem (HTP) because its nine hexagons resemble a tortoise shell. We call the sum of numbers in a hexagon in Jisuguimundo a magic sum, and show that the magic sum of hexagonal tortoise problem of order 2 varies 40 through 62 exactly and that of hexagonal tortoise problem of order 3 varies 77 through 109 exactly. We also find all of the possible solutions for hexagonal tortoise problem of oder 2.

The Jumping Knight and Other (Super) Edge-Magic Constructions

Let G be a graph of order p and size q with loops allowed. A bijective function \({f:V(G)\cup E(G)\rightarrow \{i\}_{i=1}^{p+q}}\) is an edge-magic labeling of G if the sum \({f(u)+f(uv)+f(v)=k}\) is independent of the choice of the edge uv. The constant k is called either the valence, the magic weight or the magic sum of the labeling f. If a graph admits an edge-magic labeling, then it is called an edge-magic graph. Furthermore, if the function f meets the extra condition that \({f(V(G))=\{i\}_{i=1}^{p}}\) then f is called a super edge-magic labeling and G is called a super edge-magic graph. A digraph D admits a labeling, namely l, if its underlying graph, und(D) admits l. In this paper, we introduce a new construction of super edge-magic labelings which are related to the classical jump of the knight on the chess game. We also use super edge-magic labelings of digraphs together with a generalization of the Kronecker product to get edge-magic labelings of some families of graphs.