Whitney graph isomorphism theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-02-18

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Line_graph


00:01:35 1 Formal definition
00:02:14 2 Example
00:03:07 3 Properties
00:03:17 3.1 Translated properties of the underlying graph
00:06:43 3.2 Whitney isomorphism theorem
00:08:15 3.3 Strongly regular and perfect line graphs
00:10:43 3.4 Other related graph families
00:12:54 4 Characterization and recognition
00:13:05 4.1 Clique partition
00:15:26 4.2 Forbidden subgraphs
00:16:36 4.3 Algorithms
00:19:49 5 Iterating the line graph operator
00:21:11 6 Generalizations
00:21:21 6.1 Medial graphs and convex polyhedra
00:23:01 6.2 Total graphs
00:23:33 6.3 Multigraphs
00:24:35 6.4 Line digraphs
00:25:09 6.5 Weighted line graphs
00:25:28 6.6 Line graphs of hypergraphs
00:25:37 7 Notes
00:27:54 8 References
00:28:05 9 External links



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.8451980510132333
Voice name: en-US-Wavenet-C


"I cannot teach anybody anything, I can only make them think."
- Socrates


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. The name line graph comes from a paper by Harary & Norman (1960) although both Whitney (1932) and Krausz (1943) 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.Hassler Whitney (1932) 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.