A line graph (also called an adjoint, conjugate, covering, derivative, derived, edge, edge-to-vertex dual, interchange, representative, or -obrazom graph) of a simple graph is obtained by associating a vertex with each edge of the graph and connecting two vertices with an edge iff the corresponding edges of have a vertex in common (Gross and Yellen 2006, p. 20).
The line graph of a directed graph is the directed graph whose vertex set corresponds to the arc set of and having an arc directed from an edge to an edge if in , the head of meets the tail of (Gross and Yellen 2006, p. 265).
Line graphs are implemented in the Wolfram Language as LineGraph[g]. Precomputed line graph identifications of many named graphs can be obtained in the Wolfram Language using GraphData[graph, "LineGraphName"].
The numbers of simple line graphs on , 2, ... vertices are 1, 2, 4, 10, 24, 63, 166, 471, 1408, ... (OEIS A132220), and the numbers of connected simple line graphs are 1, 1, 2, 5, 12, 30, 79, 227, ... (OEIS A003089).
The following table summarizes some named graphs and their corresponding line graphs.
|claw graph ||triangle graph |
|complete bipartite graph ||prism graph |
|cubical graph||cuboctahedral graph|
|cycle graph ||cycle graph |
|dodecahedral graph||icosidodecahedral graph|
|path graph ||singleton graph |
|path graph ||path graph |
|square graph ||square graph |
|star graph ||tetrahedral graph |
|star graph ||pentatope graph |
|star graph ||complete graph |
|tetrahedral graph ||octahedral graph|
|triangle graph ||triangle graph |
Line graphs are claw-free.
The line graph of a graph with nodes, edges, and vertex degrees contains nodes and
edges (Skiena 1990, p. 137). The incidence matrix of a graph and adjacency matrix of its line graph are related by
where is the identity matrix (Skiena 1990, p. 136).
Krausz (1943) proved that a solution exists for a simple graph iff decomposes into complete subgraphs with each vertex of appearing in at most two members of the decomposition. This theorem, however, is not useful for implementation of an efficient algorithm because of the possibly large number of decompositions involved (West 2000, p. 280).
van Rooij and Wilf (1965) shows that a solution to exists for a simple graph iff is claw-free and no induced diamond graph of has two odd triangles. Here, a triangular subgraph is said to be even if the neighborhood and vertex set intersect in an odd number of points for some and even if and intersect in an even number of points for every (West 2000, p. 281).
A simple graph is a line graph of some simple graph iff if does not contain any of the above nine graphs as an induced subgraph (van Rooij and Wilf 1965; Beineke 1968; Skiena 1990, p. 138; Harary 1994, pp. 74-75; West 2000, p. 282; Gross and Yellen 2006, p. 405). This statement is sometimes known as the Beineke theorem. These nine graphs are implemented in the Wolfram Language as GraphData["Beineke"]. Of the nine, one has four nodes (the claw graph = star graph = complete bipartite graph ), two have five nodes, and six have six nodes (including the wheel graph ).
A graph with minimum degree at least 5 is a line graph iff it does not contain any of the above six graphs as an induced subgraph (Metelsky and Tyshkevich 1997). These six graphs are implemented in the Wolfram Language as GraphData["Metelsky"].
A graph is not a line graph if the smallest element of its graph spectrum is less than (Van Mieghem, 2010, Liu et al. 2010).
Whitney (1932) showed that, with the exception of and , any two connected graphs with isomorphic line graphs are isomorphic (Skiena 1990, p. 138).
Lehot (1974) gave a linear time algorithm that reconstructs the original graph from its line graph. Liu et al. (2010) give an algorithm for reconstructing the original graph from its line graph, where is the number of vertices in the line graph. This algorithm is more time efficient than the efficient algorithm of Roussopoulos (1973).
The line graph of an Eulerian graph is both Eulerian and Hamiltonian (Skiena 1990, p. 138). More information about cycles of line graphs is given by Harary and Nash-Williams (1965) and Chartrand (1968).
Taking the line graph twice does not return the original graph unless the line graph of a graph is isomorphic to itself. In fact, The only connected graph that is isomorphic to its line graph is a cycle graph for (Skiena 1990, p. 137). Graph unions of cycle graphs (e.g., , , etc.) are also isomorphic to their line graphs, so the graphs that are isomorphic to their line graphs are the regular graphs of degree 2, and the total numbers of not-necessarily connected simple graphs that are isomorphic to their lines graphs are given by the number of partitions of their vertex count having smallest part , given for , 2, ... by 0, 0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 17, ... (OEIS A026796), the first few of which are illustrated above.