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Write $G=G^{(0)}(\tau -{\tau }^{\prime })$ and $g_0=G^{(0)}(0)$. With two identical degree-$n$ vertices, let $r$ lines join them. The remaining $n-r$ half-edges on each vertex form self-loops. Both $n$ and $r$ are even. The number of Wick contractions issince one chooses and matches the joining half-edges and then pairs the rest independently. Dividing by the expansion denominator $2!(n!{)}^2$ gives every symmetry factor weight without guessing graph multiplicities.

For two quartic vertices, $r=0,2,4$ givesThe graphs are respectively the product of two first-order two-loop graphs; two vertices joined by two lines with one self-loop on each; and two vertices joined by four lines with no self-loops.

For two sextic vertices, $r=0,2,4,6$ givesThe corresponding numbers of self-loops on each vertex are $3,2,1,0$. All seven second-order graphs are drawn in solution; in each series the $r=0$ graph is disconnected and the others are connected.

For completeness, a mixed quartic-sextic pair, not needed for the two pure second-order coefficients, contributesTaking $\log Z$ removes the products of first-order graphs, that is, the $r=0$ terms. The requested partition function retains them.