Graphs arising from the dual Steenrod algebra

We extend Wood's graph theoretic interpretation of certain quotients of the mod $2$ dual Steenrod algebra to quotients of the mod $p$ dual Steenrod algebra where $p$ is an odd prime and to quotients of the $C_2$-equivariant dual Steenrod algebra. We establish connectedness criteria for graphs associated to monomials in these algebra quotients and investigate questions about trees and Hamilton cycles in these settings. We also give graph theoretic interpretations of algebraic structures such as the coproduct and antipode arising from the Hopf algebra structure on the mod $p$ dual Steenrod algebra and the Hopf algebroid structure of the $C_2$-equivariant dual Steenrod algebra.