Antimagic Total labeling of Disjoint Union of Disconnected Graph

Abstract

A network topology (either communication network in general or a network in a computer) can be modeled as a graph or a directed graph (digraph, for short), where each processing element is represented by a vertex and the connection between two processing elements is represented by an edge (or, in the case of a digraph, by a directed arc). The number of vertices is called the order of the graph or digraph. The number of connections incident to a vertex is called the degree of the vertex. If the connections are one way only then we distinguish between in-coming and out-going connections and we speak of the in-degree and the out-degree of a vertex. The distance between two vertices is the length of the shortest path, measured by the number of edges or arcs that need to be traversed in order to reach one vertex from another vertex. In either case, the largest distance between any two vertices, called the diameter of the graph or digraph, represents the maximum data communication delay in a communication network.

By this terminology, there are a wide range of study of graph theory among mathematicians, even more when it is related to the recent digital and internet technology. A dynamical communication, flexible and massive (a huge number of elements which should be connected) is the main requirement of this network technology development. The complexity of the network will increase dramatically if the number of elements (or computer) that are involved in the network increases, especially if the number of connections that are connected to a point is also getting larger, then the network with efficient and high-speed, reliable, a good modularity, a good fault tolerance and low vulnerability will always be a major concern in designing the topology of network. One of the important efforts that can be done is to do labeling of the network topology. In other word we can do a graph labeling.

Furthermore, with an application of network technology in all aspects of modern society, it leads to the disconnected topological networks. However, obtaining a national or international publication related to the labeling for disconnected graph is still relatively few, see Gallian Dynamic Survey of Labeling. Practically it has been begun since 2004. Therefore, in this book we will present a new result for family of disconnected graph labeling such as Cycle, Path, Caterpillar, Complete n-partite Graph, Star , Crown, Triangular Lader, Generalized Petersen Graph, Banana Tree, Firecracker, Graph Lobster, Generalized web, Graph E, Diamond Ladder, Mountain Graph, Triangluar Book, Cycles non isomorphic with a chord, and stair.

This book presents some collections of new effort of studying super edge antimagic of disconnected graph in which it contains a new results as well as an open problem that can be used as references to other researchers in conducting research in labeling. This book was inspired by a book written by Martin Baca and Mirka Miler with a title of Super Edge Antimagic Labeling. They published that book in America but it was not focused on studying a disconnected graph, whereas in this book we only discus a disconnected graph.

The author is now carrying on going project concerning a labeling of disconnected graph family so that in subsequent editions of this book will present some more result of disconnected graph labeling and continuously update it every year