The tools of graph theory can be used to investigate the structure
imposed on the integers by various relations. Here we investigate two
kinds of graphs. The first, a square product graph, takes for its vertices
the integers 1 through n, and draws edges between numbers whose product
is a square. The second, a square product graph, has the same vertex set,
and draws edges between numbers whose sum is a square.
We investigate the structure of these graphs. For square product
graphs, we provide a rather complete characterization of their structure as
a union of disjoint complete graphs. For square sum graphs, we investigate
some properties such as degrees of vertices, connectedness, hamiltonicity,
Trent, Lee, "Structure of Number Theoretic Graphs" (2022). Mathematical Sciences Technical Reports (MSTR). 179.