[proofplan] We translate the graphical statement into the corresponding statement for the graphical matroid $M(G)$. In a graphical matroid, graph deletion and graph contraction agree with matroid deletion and matroid contraction, and loops and bridges become matroid loops and coloops. The result then follows from the deletion-contraction recurrence for characteristic polynomials of matroids, applied to $M(G)$ and then translated back to lattices of flats of the resulting graphical matroids. [/proofplan] [step:Pass from the graph to its graphical matroid] Let $M:=M(G)$ be the graphical matroid of $G$ on ground set $E$. By definition, the independent sets of $M$ are the forests in $G$, and the rank of a set $A\subset E$ is the size of a maximal forest contained in the spanning subgraph $(V,A)$. The lattice $L(G)$ is, by hypothesis and notation, the lattice of flats of $M$, so $\chi_{L(G)}(q)=\chi_M(q)$. For an edge $e\in E$, graph deletion and contraction induce the corresponding matroid minors: $M(G\setminus e)=M\setminus e$ and $M(G/e)=M/e$. Here $G/e$ is allowed to have loops and parallel edges, which is compatible with the graphical matroid construction. [guided] We first replace the graph-theoretic notation by matroid notation, because the [characteristic polynomial](/page/Characteristic%20Polynomial) recurrence is fundamentally a matroid recurrence. Define \begin{align*} M:=M(G). \end{align*} This is the graphical matroid of $G$ with ground set $E$. Its independent sets are exactly the edge sets that contain no cycle in $G$, equivalently the forests of $G$. The symbol $L(G)$ denotes the lattice of flats of this matroid $M(G)$. Therefore the characteristic polynomial appearing in the statement is not being treated as the chromatic polynomial of $G$; it is the characteristic polynomial of the lattice of flats of $M(G)$. With this convention, \begin{align*} \chi_{L(G)}(q)=\chi_M(q). \end{align*} Now consider the edge $e\in E$. Deleting $e$ from the graph deletes the same element from the graphical matroid: \begin{align*} M(G\setminus e)=M\setminus e. \end{align*} Contracting $e$ in the graph contracts the corresponding matroid element: \begin{align*} M(G/e)=M/e. \end{align*} The graph $G/e$ may acquire loops or parallel edges, but graphical matroids are defined for such finite graphs, so the notation remains valid. This is the bridge between the graph statement and the matroid recurrence. [/guided] [/step] [step:Apply deletion-contraction in the nondegenerate case] Assume that $e$ is neither a loop nor a bridge of $G$. In the graphical matroid $M(G)$, loops of $G$ are precisely loops of $M$, and bridges of $G$ are precisely coloops of $M$. Hence $e$ is neither a loop nor a coloop of $M$. By [citetheorem:8114] applied to the matroid $M$ and the element $e$, we obtain $\chi_M(q)=\chi_{M\setminus e}(q)-\chi_{M/e}(q)$. Using the identifications from the previous step, $\chi_{M\setminus e}(q)=\chi_{L(G\setminus e)}(q)$ and $\chi_{M/e}(q)=\chi_{L(G/e)}(q)$. Substituting these equalities gives $\chi_{L(G)}(q)=\chi_{L(G\setminus e)}(q)-\chi_{L(G/e)}(q)$. [/step] [step:Record the loop and bridge cases] If $e$ is a loop of $G$, then $e$ is a loop of $M(G)$. By the loop case of [citetheorem:8114], $\chi_{L(G)}(q)=\chi_M(q)=0$. If $e$ is a bridge of $G$, then $e$ is a coloop of $M(G)$. By the coloop case of [citetheorem:8114], $\chi_M(q)=(q-1)\chi_{M/e}(q)$. Using $M/e=M(G/e)$, this becomes $\chi_{L(G)}(q)=(q-1)\chi_{L(G/e)}(q)$. This proves both the deletion-contraction identity and the two degenerate cases. [/step]