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Directed acyclic graphs whose nodes are all the divisors of a positiveinteger $n$ and arcs $(a,b)$ defined by $a$ divides $b$ are considered.Fourteen graph invariants such as order, size, and the number of paths areinvestigated for two classic graphs, the Hasse diagram $G^H(n)$ and itstransitive closure $G^T(n)$ derived from the divides relation partial order.Concise formulae and algorithms are devised for these graph invariants andseveral important properties of these graphs are formally proven. Integersequences of these invariants in natural order by $n$ are computed and severalnew sequences are identified by comparing them to existing sequences in theOn-Line Encyclopedia of Integer Sequences. These new and existing integersequences 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, twoconventional orders of prime signatures, namely the graded colexicographic andthe canonical orders are considered and additional new integer sequences arediscovered.

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