  A 4-point Feynman diagram in scalar $\phi^4$ theory is represented by a graph
$G$ which is obtained from a connected 4-regular graph by deleting a vertex.
The associated Feynman integral gives a quantity called the period of $G$ which
is invariant under a number of meaningful graph operations - namely, planar
duality, the Schnetz twist, and it also does not depend on the choice of vertex
which was deleted to form $G$.
  In this article we study a graph invariant we call the graph permanent, which
was implicitly introduced in a paper by Alon, Linial and Meshulam. The graph
permanent applies to any graph $G = (V,E)$ for which $|E|$ is a multiple of
$|V| - 1$ (so in particular to graphs obtained from a 4-regular graph by
removing a vertex). We prove that the graph permanent, like the period, is
invariant under planar duality and the Schnetz twist when these are valid
operations, and we show that when $G$ is obtained from a $2k$-regular graph by
deleting a vertex, the graph permanent does not depend on the choice of deleted
vertex.
