Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures

Abstract

Directed acyclic graphs whose nodes are all the divisors of a positive integer $n$ and arcs $(a,b)$ defined by $a$ divides $b$ are considered. Fourteen graph invariants such as order, size, and the number of paths are investigated for two classic graphs, the Hasse diagram $G^H(n)$ and its transitive closure $G^T(n)$ derived from the divides relation partial order. Concise formulae and algorithms are devised for these graph invariants and several important properties of these graphs are formally proven. Integer sequences of these invariants in natural order by $n$ are computed and several new sequences are identified by comparing them to existing sequences in the On-Line Encyclopedia of Integer Sequences. These new and existing integer sequences are interpreted from the graph theory point of view. Both $G^H(n)$ and $G^T(n)$ are characterized by the prime signature of $n$. Hence, two conventional orders of prime signatures, namely the graded colexicographic and the canonical orders are considered and additional new integer sequences are discovered.