Further investigations into the graph theory of $\phi^4$-periods and the $c_2$ invariant
A Feynman period is a particular residue of a scalar Feynman integral which is both physically and number theoretically interesting. Two ways in which the graph theory of the underlying Feynman graph can illuminate the Feynman period are via graph operations which are period invariant and other graph quantities which predict aspects of the Feynman period, one notable example is known as the $c_2$ invariant. We give results and computations in both these directions, proving a new period identity and computing its consequences up to 11 loops in $\phi^4$-theory, proving a $c_2$ invariant identity, and giving the results of a computational investigation of $c_2$ invariants at 11 loops.