On the Magic Space of Locally Finite Graphs

Abstract

A labelling of a graph over a field \(\mathbb{F}\) is a mapping of the edge set of the graph into \(\mathbb{F}\). A labelling is called magic if for any vertex, the sum of the labels of all the edges incident to it is the same. The class of all such labellings forms a vector space over \(\mathbb{F}\) and is called the magic space of the graph. For finite graphs, the dimensional structure of the magic space is well known. In this paper, we give the existence of magic labellings and discuss the dimensional structure of the magic space of locally finite graphs. In particular, for a class of locally finite graphs, we give an explicit basis of the magic space.