  Feynman diagrams in $\phi^4$ theory have as their underlying structure
4-regular graphs. In particular, any 4-point $\phi^4$ graph can be uniquely
derived from a 4-regular graph by deleting a vertex. The Feynman period is a
simplified version of the Feynman integral, and is of special interest, as it
maintains much of the important number theoretic information from the Feynman
integral. It is also of structural interest, as it is known to be preserved by
a number of graph theoretic operations. In particular, the vertex deleted in
constructing the 4-point graph does not affect the Feynman period, and it is
invariant under planar duality and the Schnetz twist, an operation that
redirects edges incident to a 4-vertex cut. Further, a 4-regular graph may be
produced by a 3-sum operation on triangles in two 4-regular graphs. The Feynman
period of this graph with a vertex deleted is equal to the product of the
Feynman periods of the two smaller graphs with one vertex deleted each. These
operations currently explain all known instances of non-isomorphic 4-point
$\phi^4$ graphs with equal periods.
  With this in mind, other graph invariants that are preserved by these
operations for 4-point $\phi^4$ graphs are of interest, as they may provide
insight into the Feynman period. In this thesis the extended graph permanent is
introduced; an infinite sequence of residues from prime order finite fields. It
is shown that this sequence is preserved by these three operations, and has a
product property. Additionally, computational techniques will be established,
and an alternate interpretation will be presented as the point count of a novel
graph polynomial.
  Further, the previously existing $c_2$ invariant and Hepp bound are examined,
two graph invariants that are conjectured to be preserved by these graph
operations. A combinatorial approach to the $c_2$ invariant is introduced.
