A SURVEY OF DISTANCE MAGIC GRAPHS

In this report, we survey results on distance magic graphs and some closely related graphs. A distance magic labeling of a graph G with magic constant k is a bijection l from the vertex set to {1, 2, . . . , n}, such that for every vertex x Σ l(y) = k,y∈NG(x) where NG(x) is the set of vertices of G adjacent to x. If the graph G has a distance magic labeling we say that G is a distance magic graph. In Chapter 1, we explore the background of distance magic graphs by introducing examples of magic squares, magic graphs, and distance magic graphs. In Chapter 2, we begin by examining some basic results on distance magic graphs. We next look at results on different graph structures including regular graphs, multipartite graphs, graph products, join graphs, and splitting graphs. We conclude with other perspectives on distance magic graphs including embedding theorems, the matrix representation of distance magic graphs, lifted magic rectangles, and distance magic constants. In Chapter 3, we study graph labelings that retain the same labels as distance magic labelings, but alter the definition in some other way. These labelings include balanced distance magic labelings, closed distance magic labelings, D-distance magic labelings, and distance antimagic labelings. In Chapter 4, we examine results on neighborhood magic labelings, group distance magic labelings, and group distance antimagic labelings. These graph labelings change the label set, but are otherwise similar to distance magic graphs. In Chapter 5, we examine some applications of distance magic and distance antimagic labeling to the fair scheduling of tournaments. In Chapter 6, we conclude with some open problems

A Complete Characterization of all Magic Constants Arising from Distance Magic Graphs

A positive integer $k$ is called a magic constant if there is a graph $G$ along with a bijective function $f$ from $V(G)$ to first $|V(G)|$ natural numbers such that the weight of the vertex $w(v) = \sum_{uv \in E}f(v) =k$ for all $v \in V$. It is known that all odd positive integers greater equal $3$ and the integer powers of $2$, $2^{t}$, $t \ge 6$ are magic constants. In this paper we characterise all positive integers which are magic constants

NOTIFICATION !!!

All the content of this special edition is retrieved from the conference proceedings published by the European Scientific Institute, ESI. http://eujournal.org/index.php/esj/pages/view/books The European Scientific Journal, ESJ, after approval from the publisher re publishes the papers in a Special edition

NOTIFICATION !!!

All the content of this special edition is retrieved from the conference proceedings published by the European Scientific Institute, ESI. http://eujournal.org/index.php/esj/pages/view/books The European Scientific Journal, ESJ, after approval from the publisher re publishes the papers in a Special edition

NOTIFICATION !!!

All the content of this special edition is retrieved from the conference proceedings published by the European Scientific Institute, ESI. http://eujournal.org/index.php/esj/pages/view/books The European Scientific Journal, ESJ, after approval from the publisher re publishes the papers in a Special edition

NOTIFICATION !!!

All the content of this special edition is retrieved from the conference proceedings published by the European Scientific Institute, ESI. http://eujournal.org/index.php/esj/pages/view/books The European Scientific Journal, ESJ, after approval from the publisher re publishes the papers in a Special edition