**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Line graph

Summary

In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. L(G) is constructed in the following way: for each edge in G, make a vertex in L(G); for every two edges in G that have a vertex in common, make an edge between their corresponding vertices in L(G).
The name line graph comes from a paper by although both and used the construction before this. Other terms used for the line graph include the covering graph, the derivative, the edge-to-vertex dual, the conjugate, the representative graph, and the θ-obrazom, as well as the edge graph, the interchange graph, the adjoint graph, and the derived graph.
proved that with one exceptional case the structure of a connected graph G can be recovered completely from its line graph. Many other properties of line graphs follow by translating the properties of the underlying graph from vertices into edges, and by Whitney's theorem the same translation can also be done in the other direction. Line graphs are claw-free, and the line graphs of bipartite graphs are perfect. Line graphs are characterized by nine forbidden subgraphs and can be recognized in linear time.
Various extensions of the concept of a line graph have been studied, including line graphs of line graphs, line graphs of multigraphs, line graphs of hypergraphs, and line graphs of weighted graphs.
Given a graph G, its line graph L(G) is a graph such that
each vertex of L(G) represents an edge of G; and
two vertices of L(G) are adjacent if and only if their corresponding edges share a common endpoint ("are incident") in G.
That is, it is the intersection graph of the edges of G, representing each edge by the set of its two endpoints.
The following figures show a graph (left, with blue vertices) and its line graph (right, with green vertices). Each vertex of the line graph is shown labeled with the pair of endpoints of the corresponding edge in the original graph.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (35)

Related people (112)

Related units (9)

Related courses (32)

Related publications (541)

Related MOOCs (13)

Forbidden graph characterization

In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidden graphs as (induced) subgraph or minor. A prototypical example of this phenomenon is Kuratowski's theorem, which states that a graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph K_5 and the complete bipartite graph K_3,3.

Edge coloring

In graph theory, a proper edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most k different colors, for a given value of k, or with the fewest possible colors.

Claw-free graph

In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph. A claw is another name for the complete bipartite graph K1,3 (that is, a star graph comprising three edges, three leaves, and a central vertex). A claw-free graph is a graph in which no induced subgraph is a claw; i.e., any subset of four vertices has other than only three edges connecting them in this pattern. Equivalently, a claw-free graph is a graph in which the neighborhood of any vertex is the complement of a triangle-free graph.

PHYS-512: Statistical physics of computation

This course covers the statistical physics approach to computer science problems ranging from graph theory and constraint satisfaction to inference and machine learning. In particular the replica and

EE-548: Audio engineering

This lecture is oriented towards the study of audio engineering, with a special focus on room acoustics applications. The learning outcomes will be the techniques for microphones and loudspeaker desig

ME-427: Networked control systems

This course offers an introduction to control systems using communication networks for interfacing sensors, actuators, controllers, and processes. Challenges due to network non-idealities and opportun

Analyse I

Le contenu de ce cours correspond à celui du cours d'Analyse I, comme il est enseigné pour les étudiantes et les étudiants de l'EPFL pendant leur premier semestre. Chaque chapitre du cours correspond

Analyse I (partie 1) : Prélude, notions de base, les nombres réels

Concepts de base de l'analyse réelle et introduction aux nombres réels.

Analyse I (partie 2) : Introduction aux nombres complexes

Introduction aux nombres complexes

Related lectures (343)

Linear Algebra: Matrices and Linear Applications

Covers matrices, linear applications, vector spaces, and bijective functions.

Linear Algebra: Matrix Representation

Explores linear applications in R² and matrix representation, including basis, operations, and geometric interpretation of transformations.

Vector Algebra: Scalar Product

Explores the scalar product of vectors, including properties and calculation methods.

We prove the non-planarity of a family of 3-regular graphs constructed from the solutions to the Markoff equation x2 + y2 + z2 = xyz modulo prime numbers greater than 7. The proof uses Euler characteristic and an enumeration of the short cycles in these gr ...

Spectral algorithms are some of the main tools in optimization and inference problems on graphs. Typically, the graph is encoded as a matrix and eigenvectors and eigenvalues of the matrix are then used to solve the given graph problem. Spectral algorithms ...

Maximilien Claude Robert Dreveton

This article focuses on spectral methods for recovering communities in temporal networks. In the case of fixed communities, spectral clustering on the simple time-aggregated graph (i.e., the weighted graph formed by the sum of the interactions over all tem ...