D-Magic Oriented Graphs

In this paper, we define D-magic labelings for oriented graphs where D is a distance set. In particular, we label the vertices of the graph with distinct integers {1,2,…,|V(G)|} in such a way that the sum of all the vertex labels that are a distance in D away from a given vertex is the same across all vertices. We give some results related to the magic constant, construct a few infinite families of D-magic graphs, and examine trees, cycles, and multipartite graphs. This definition grew out of the definition of D-magic (undirected) graphs. This paper explores some of the symmetries we see between the undirected and directed version of D-magic labelings.

Some properties of magSome Properties of Magic Squares of Distinct Squares and Cubesic squares

Magic squares is n×n matrix with positive integer entries as well as the sum of rows, columnsand mains diagonal have the same magic constant, one of the most oldest magic square wasdiscovered in china. In this paper the history of magic square is displayed and some definitionof its kind is given the prove of two theorems about properties of magic square is introduced.

A Modification of the Fast Inverse Square Root Algorithm

We present a new algorithm for the approximate evaluation of the inverse square root for single-precision floating-point numbers. This is a modification of the famous fast inverse square root code. We use the same “magic constant” to compute the seed solution, but then, we apply Newton–Raphson corrections with modified coefficients. As compared to the original fast inverse square root code, the new algorithm is two-times more accurate in the case of one Newton–Raphson correction and almost seven-times more accurate in the case of two corrections. We discuss relative errors within our analytical approach and perform numerical tests of our algorithm for all numbers of the type float.

Frenicle’s 880 Magic Squares

This chapter discusses magic squares. A magic square of order n is an arrangement of the numbers from 1 to n 2 in an n × n array so that the two diagonals and all the rows and columns have the same sum. This sum is called the magic constant. Bernard Frenicle de Bessy's work on magic squares appears in two papers published in the book Divers ouvrages de mathematique et de physique par Messieurs de l'Academie Royale des Sciences. In his first paper, “Des Quarrez ou Tables Magiques,” Frenicle quotes a rule for constructing magic squares of odd order. However, Frenicle is more famous for his second paper, “Table Generale des Quarrez de Quatres,” in which he enumerates the 880 magic squares of order four. His enumeration has been repeated many times. These later enumerations they have confirmed the remarkable fact that he was correct.

Multi-Wheels are Cycle Anti-Supermagic

A graph G(V,E) has an H-covering if every edge in E belongs to a subgraph of G isomorphic to H. Suppose G admits an H-covering. An H-magic labeling is a total labeling λ from V(G)∪E(G) onto the integers {1, 2, …, |V(G)∪E(G)|} with the property that, for every subgraph A of G isomorphic to H there is a positive integer c such that <inline-formula> <mml:math display="block"> <mml:mo>∑</mml:mo><mml:mi>A</mml:mi><mml:mo>=</mml:mo><mml:msub> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>v</mml:mi><mml:mo>∈</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> <mml:mi>λ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>e</mml:mi><mml:mo>∈</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> <mml:mi>λ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>e</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mo>.</mml:mo> </mml:math> </inline-formula> A graph that admits such a labeling is called H-magic. In addition, if <inline-formula> <mml:math display="block"> <mml:msub> <mml:mrow><mml:mo>{</mml:mo> <mml:mrow> <mml:mi>λ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>V</mml:mi><mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mo>}</mml:mo></mml:mrow> <mml:mrow> <mml:mi>v</mml:mi><mml:mi>e</mml:mi><mml:mi>V</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo> <mml:mrow> <mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>...</mml:mn><mml:mo>,</mml:mo><mml:mo>|</mml:mo><mml:mi>V</mml:mi><mml:mo>|</mml:mo> </mml:mrow> <mml:mo>}</mml:mo></mml:mrow><mml:mo>,</mml:mo> </mml:math> </inline-formula> then the graph is called H-supermagic. Moreover a graph is said to be H-(a,d) antimagic if the magic constant for an arithmetics progression with initial value A and a common difference d. In this paper we formulate cycle c3 − (a, d) anti-supermagic labelings for the Multi-Wheels graph, supermagic labelings for disjoint union of isomorphic copies of Multi-Wheels graph and cycle (a, d)-anti-supermagic labelings for Web graph.

Zk-Magic labeling of subdivision graphs

For any nontrivial abelian group [Formula: see text] under addition a graph [Formula: see text] is said to be [Formula: see text]-magic if there exists a labeling [Formula: see text] such that the vertex labeling [Formula: see text] defined as [Formula: see text] taken over all edges [Formula: see text] incident at [Formula: see text] is a constant. An [Formula: see text]-magic graph [Formula: see text] is said to be [Formula: see text]-magic graph if the group [Formula: see text] is [Formula: see text] the group of integers modulo [Formula: see text]. These [Formula: see text]-magic graphs are referred to as [Formula: see text]-magic graphs. In this paper, we prove that the graphs such as subdivision of ladder, triangular ladder, shadow, total, flower, generalized prism, [Formula: see text]-snake, lotus inside a circle, square, gear, closed helm and antiprism are [Formula: see text]-magic graphs. Also we prove that if [Formula: see text] be [Formula: see text]-magic graphs with magic constant zero then [Formula: see text] is also [Formula: see text]-magic.

Note on group distance magic complete bipartite graphs

A Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection ℓ from V to an Abelian group Γ of order n such that the weight $$w(x) = \sum\nolimits_{y \in N_G (x)} {\ell (y)}$$ of every vertex x ∈ V is equal to the same element µ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|.In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ℤp-distance magic. Moreover we show that if p ≡ 2 (mod 4) and k is even, then there does not exist a group Γ of order p such that there exists a Γ-distance labeling for a k-partite complete graph of order p. We also prove that K m,n is a group distance magic graph if and only if n + m ≢ 2 (mod 4).