In Graph Theory , (also called '''parallel edges''' or a '''multi-edge'''), are two or more Edges that are Incident to the same two Vertices . A Simple Graph has no multiple edges.
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 is often called a Multigraph . For example, see Balakrishnan, p. 1, and Gross (2003), p. 4, Zwillinger, p. 220.
- 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 loops and multiple edges.For example, see. Bollobas, p. 7, Diestel, p. 25, and Harary, p. 10.
Multiple edges are, for example, useful in the consideration of Electrical Network s, from a graph theoretical point of view. Bollobas, pp. 39, 40.
A Planar Graph remains planar if an edge is added between two vertices already joined by an edge, thus adding multiple edges preserves planarity.Gross (1998), p. 308.
- Balakrishnan, V. K.; ''Graph Theory'', McGraw-Hill; 1 edition (February 1, 1997). ISBN 0-07-005489-4.
- Bollobas, Bela; ''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.
|
