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

On Structural and Spectral Properties of Distance Magic Graphs

A graph $G=(V,E)$ is said to be distance magic if there is a bijection $f$ from a vertex set of $G$ to the first $|V(G)|$ natural numbers such that for each vertex $v$, its weight given by $\sum_{u \in N(v)}f(u)$ is constant, where $N(v)$ is an open neighborhood of a vertex $v$. In this paper, we introduce the concept of $p$-distance magic labeling and establish the necessary and sufficient condition for a graph to be distance magic. Additionally, we introduce necessary and sufficient conditions for a connected regular graph to exhibit distance magic properties in terms of the eigenvalues of its adjacency and Laplacian matrices. Furthermore, we study the spectra of distance magic graphs, focusing on singular distance magic graphs. Also, we show that the number of distance magic labelings of a graph is, at most, the size of its automorphism group

Graph Labelings and Tournament Scheduling

University of Minnesota M.S. thesis. May 2015. Major: Applied and Computational Mathematics. Advisor: Dalibor Froncek. 1 computer file (PDF); viii, 55 pages.During my research I studied and became familiar with distance magic and distance antimagic labelings and their relation to tournament scheduling. Roughly speaking, the relation is as follows. Let the vertices on the graph represent teams in a tournament, and let an edge between two vertices a and b represent that team a will play team b in the tournament. Further, suppose we can rank the teams based on previous games, say, the preceding season. These integer rankings become labels for the vertices. Of particular interest were handicap tournaments, that is, tournaments designed to give each team a more balanced chance of winning