Multiple edges
Today we are going to delve into the world of Multiple edges, a topic that has been the subject of interest and debate in different areas over time. Multiple edges has aroused the curiosity and fascination of many people due to its importance in today's society. Throughout history, Multiple edges has played a fundamental role in different contexts, from politics and culture to science and technology. Throughout this article, we will explore the importance of Multiple edges, as well as its impact on society at large. Additionally, we will look at how Multiple edges has evolved over time and how it has influenced our lives in ways we may not have considered before. Get ready to enter the fascinating world of Multiple edges!
In graph theory, multiple edges (also called parallel edges or a multi-edge), are, in an undirected graph, two or more edges that are incident to the same two vertices, or in a directed graph, two or more edges with both the same tail vertex and the same head vertex. A simple graph has no multiple edges and no loops.
Depending on the context, a graph may be defined so as to either allow or disallow the presence of multiple edges (often in concert with allowing or disallowing loops):
- Where graphs are defined so as to allow multiple edges and loops, a graph without loops or multiple edges is often distinguished from other graphs by calling it a simple graph.[1]
- Where graphs are defined so as to disallow multiple edges and loops, a multigraph or a pseudograph is often defined to mean a "graph" which can have multiple edges.[2]
Multiple edges are, for example, useful in the consideration of electrical networks, from a graph theoretical point of view.[3] Additionally, they constitute the core differentiating feature of multidimensional networks.
A planar graph remains planar if an edge is added between two vertices already joined by an edge; thus, adding multiple edges preserves planarity.[4]
A dipole graph is a graph with two vertices, in which all edges are parallel to each other.
Notes
References
- Balakrishnan, V. K.; Graph Theory, McGraw-Hill; 1 edition (February 1, 1997). ISBN 0-07-005489-4.
- Bollobás, Béla; Modern Graph Theory, Springer; 1st edition (August 12, 2002). ISBN 0-387-98488-7.
- Diestel, Reinhard; Graph Theory, Springer; 2nd edition (February 18, 2000). ISBN 0-387-98976-5.
- Gross, Jonathon L, and Yellen, Jay; Graph Theory and Its Applications, CRC Press (December 30, 1998). ISBN 0-8493-3982-0.
- Gross, Jonathon L, and Yellen, Jay; (eds); Handbook of Graph Theory. CRC (December 29, 2003). ISBN 1-58488-090-2.
- Zwillinger, Daniel; CRC Standard Mathematical Tables and Formulae, Chapman & Hall/CRC; 31st edition (November 27, 2002). ISBN 1-58488-291-3.