Magic and antimagic labeling of graphs

"A bijection mapping that assigns natural numbers to vertices and/or edges of a graph is called a labeling. In this thesis, we consider graph labelings that have weights associated with each edge and/or vertex. If all the vertex weights (respectively, edge weights) have the same value then the labeling is called magic. If the weight is different for every vertex (respectively, every edge) then we called the labeling antimagic. In this thesis we introduce some variations of magic and antimagic labelings and discuss their properties and provide corresponding labeling schemes. There are two main parts in this thesis. One main part is on vertex labeling and the other main part is on edge labeling."Doctor of Philosoph

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Structural properties and labeling of graphs

The complexity in building massive scale parallel processing systems has re- sulted in a growing interest in the study of interconnection networks design. Network design affects the performance, cost, scalability, and availability of parallel computers. Therefore, discovering a good structure of the network is one of the basic issues. From modeling point of view, the structure of networks can be naturally stud- ied in terms of graph theory. Several common desirable features of networks, such as large number of processing elements, good throughput, short data com- munication delay, modularity, good fault tolerance and diameter vulnerability correspond to properties of the underlying graphs of networks, including large number of vertices, small diameter, high connectivity and overall balance (or regularity) of the graph or digraph. The first part of this thesis deals with the issue of interconnection networks ad- dressing system. From graph theory point of view, this issue is mainly related to a graph labeling. We investigate a special family of graph labeling, namely antimagic labeling of a class of disconnected graphs. We present new results in super (a; d)-edge antimagic total labeling for disjoint union of multiple copies of special families of graphs. The second part of this thesis deals with the issue of regularity of digraphs with the number of vertices close to the upper bound, called the Moore bound, which is unobtainable for most values of out-degree and diameter. Regularity of the underlying graph of a network is often considered to be essential since the flow of messages and exchange of data between processing elements will be on average faster if there is a similar number of interconnections coming in and going out of each processing element. This means that the in-degree and out-degree of each processing element must be the same or almost the same. Our new results show that digraphs of order two less than Moore bound are either diregular or almost diregular.Doctor of Philosoph

Magic and supermagic dense bipartite graphs

A graph is called magic (supermagic) if it admits a labelling of the edges by pairwise different (and consecutive) positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In the paper we prove that any balanced bipartite graph with minimum degree greater than |V(G)|/4 ≥ 2 is magic. A similar result is presented for supermagic regular bipartite graphs