In the mathematical theory of matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the forests in a given finite undirected graph. The dual matroids of graphic matroids are called co-graphic matroids or bond matroids. A matroid that is both graphic and co-graphic is sometimes called a planar matroid (but this should not be confused with matroids of rank 3, which generalize planar point configurations); these are exactly the graphic matroids formed from planar graphs.

A matroid may be defined as a family of finite sets (called the "independent sets" of the matroid) that is closed under subsets and that satisfies the "exchange property": if sets ${\displaystyle A}$ and ${\displaystyle B}$ are both independent, and ${\displaystyle A}$ is larger than ${\displaystyle B}$, then there is an element ${\displaystyle x\in A\setminus B}$ such that ${\displaystyle B\cup \{x\}}$ remains independent. If ${\displaystyle G}$ is an undirected graph, and ${\displaystyle F}$ is the family of sets of edges that form forests in ${\displaystyle G}$, then ${\displaystyle F}$ is clearly closed under subsets (removing edges from a forest leaves another forest). It also satisfies the exchange property: if ${\displaystyle A}$ and ${\displaystyle B}$ are both forests, and ${\displaystyle A}$ has more edges than ${\displaystyle B}$, then it has fewer connected components, so by the pigeonhole principle there is a component ${\displaystyle C}$ of ${\displaystyle A}$ that contains vertices from two or more components of ${\displaystyle B}$. Along any path in ${\displaystyle C}$ from a vertex in one component of ${\displaystyle B}$ to a vertex of another component, there must be an edge with endpoints in two components, and this edge may be added to ${\displaystyle B}$ to produce a forest with more edges. Thus, ${\displaystyle F}$ forms the independent sets of a matroid, called the graphic matroid of ${\displaystyle G}$ or ${\displaystyle M(G)}$. More generally, a matroid is called graphic whenever it is isomorphic to the graphic matroid of a graph, regardless of whether its elements are themselves edges in a graph.

The bases of a graphic matroid ${\displaystyle M(G)}$ are the full spanning forests of ${\displaystyle G}$, and the circuits of ${\displaystyle M(G)}$ are the simple cycles of ${\displaystyle G}$. The rank in ${\displaystyle M(G)}$ of a set ${\displaystyle X}$ of edges of a graph ${\displaystyle G}$ is ${\displaystyle r(X)=n-c}$ where ${\displaystyle n}$ is the number of vertices in the subgraph formed by the edges in ${\displaystyle X}$ and ${\displaystyle c}$ is the number of connected components of the same subgraph. The corank of the graphic matroid is known as the circuit rank or cyclomatic number.

The closure ${\displaystyle \operatorname {cl} (S)}$ of a set ${\displaystyle S}$ of edges in ${\displaystyle M(G)}$ is a flat consisting of the edges that are not independent of ${\displaystyle S}$ (that is, the edges whose endpoints are connected to each other by a path in ${\displaystyle S}$). This flat may be identified with the partition of the vertices of ${\displaystyle G}$ into the connected components of the subgraph formed by ${\displaystyle S}$: Every set of edges having the same closure as ${\displaystyle S}$ gives rise to the same partition of the vertices, and ${\displaystyle \operatorname {cl} (S)}$ may be recovered from the partition of the vertices, as it consists of the edges whose endpoints both belong to the same set in the partition. In the lattice of flats of this matroid, there is an order relation ${\displaystyle x\leq y}$ whenever the partition corresponding to flat ${\displaystyle x}$ is a refinement of the partition corresponding to flat ${\displaystyle y}$.

In this aspect of graphic matroids, the graphic matroid for a complete graph ${\displaystyle K_{n}}$ is particularly important, because it allows every possible partition of the vertex set to be formed as the set of connected components of some subgraph. Thus, the lattice of flats of the graphic matroid of ${\displaystyle K_{n}}$ is naturally isomorphic to the lattice of partitions of an ${\displaystyle n}$-element set. Since the lattices of flats of matroids are exactly the geometric lattices, this implies that the lattice of partitions is also geometric.

The graphic matroid of a graph ${\displaystyle G}$ can be defined as the column matroid of any oriented incidence matrix of ${\displaystyle G}$. Such a matrix has one row for each vertex, and one column for each edge. The column for edge ${\displaystyle e}$ has ${\displaystyle +1}$ in the row for one endpoint, ${\displaystyle -1}$ in the row for the other endpoint, and ${\displaystyle 0}$ elsewhere; the choice of which endpoint to give which sign is arbitrary. The column matroid of this matrix has as its independent sets the linearly independent subsets of columns.

If a set of edges contains a cycle, then the corresponding columns (multiplied by ${\displaystyle -1}$ if necessary to reorient the edges consistently around the cycle) sum to zero, and is not independent. Conversely, if a set of edges forms a forest, then by repeatedly removing leaves from this forest it can be shown by induction that the corresponding set of columns is independent. Therefore, the column matrix is isomorphic to ${\displaystyle M(G)}$.

This method of representing graphic matroids works regardless of the field over which the incidence is defined. Therefore, graphic matroids form a subset of the regular matroids, matroids that have representations over all possible fields.