We consider graphs on monomials in n variables of a fixed degree d where two monomials are adjacent if and only if their least common multiple has degree d+1. We find that when n = 3 and d is divisible by 3 as well as when n = 4 and d is even that these graphs have a unique maximum independent set. Domination in these graphs is also considered, and we conjecture that there is equality of the domination number and independent domination number in all cases.

We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these combinatorial Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their $f$-vectors. We also use characters to give a generalization of Stanley's $(-1)$-color theorem. A $q$-analogue version of this family of characters is also studied.