structural:reformatting
Other related graph families

All line graphs are claw-free graphs, graphs without an induced subgraph in the form of a three-leaf tree. As with claw-free graphs more generally, every connected line graph L (G) with an even number of edges has a perfect matching; equivalently, this means that if the underlying graph G has an even number of edges, its edges can be partitioned into two-edge paths.

The line graphs of trees are exactly the claw-free block graphs. These graphs have been used to solve a problem in extremal graph theory, of constructing a graph with a given number of edges and vertices whose largest tree induced as a subgraph is as small as possible.

All eigenvalues of the adjacency matrix A of a line graph are at least −2. The reason for this is that A can be written as A = J T J − 2 I {\displaystyle A=J^{\mathsf {T}}J-2I}, where J is the signless incidence matrix of the pre-line graph and I is the identity. In particular, A + 2 I is the Gramian matrix of a system of vectors: all graphs with this property have been called generalized line graphs.