Harmonious Labeling of Infinite Graphs
In graph theory, graphs are mathematical structures containing a set of vertices and a set of edges that represent the pairwise relationship of objects. A graph can be labeled with a function that assigns a number to each vertex. A harmonious labeling of a graph G assigns positive numbers to the vertices such that the sum of each adjacent vertex label is distinct modulo the number of edges in G. In our research we expanded the definition of harmonious labelings to apply to infinite graphs, and investigated which infinite graphs are harmonious by our definition. We also defined and investigated a new type of labeling for both finite and infinite graphs, locally harmonious labeling.