This submission defines five parameters of finite simple undirected graphs: maximum matching number, twin-cover number, neighborhood diversity, degeneracy, and arboricity. Disconnected graphs are included, and all five parameters assign value zero to the empty graph.

The parameters are defined through matchings, twin-covers, neighborhood partitions, degree conditions on induced subgraphs, and partitions of the edge set into forests, respectively. Each parameter has its own definition concept.

Five further definitions express functional boundedness, functional equivalence, absence of a functional bound, strict functional boundedness, and incomparability on all finite simple graphs. A theorem, with a Lean proof, identifies mutual nondecreasing functional bounds with the existing Lax notion of functional equivalence. No comparisons between the five concrete parameters are asserted.

A second theorem establishes that exactly one of four relations holds for every parameter pair: strict functional boundedness in either direction, functional equivalence, or incomparability.

A third theorem proves that functional equivalence is an equivalence relation. A fourth proves that strict functional boundedness is a strict partial order. A fifth proves that incomparability is symmetric and irreflexive.