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For any connected multigraph $G=(V,E)$ and any $M\subseteq E$, if $M$ inducesan acyclic subgraph of $G$ and removing all edges in $M$ yields a subgraph of$G$ whose components are complete graphs, a formula for $\tau_G(M)$ isobtained, where $\tau_G(M)$ is the number of spanning trees in $G$ whichcontain all edges in $M$. Applying this result, we can easily obtain a formulafor the number of spanning trees in the line graph or the middle graph of anarbitrary graph. Applying this result, we also show that for any connectedgraph $G$ with a clique $U$ which is a cut-set of $G$, the number of spanningtrees in $G$ has a factorization which is analogous to a property of thechromatic polynomial of $G$.

Formulas counting spanning trees in line graphs and their extensions