Content deleted Content added
Citation bot (talk | contribs) Add: url. | You can use this bot yourself. Report bugs here. | Suggested by SemperIocundus | via #UCB_webform |
source triangular cactus |
||
Line 11:
===Gluing and products===
The [[friendship graph]]s, graphs formed by gluing together a collection of triangles at a single shared vertex, are locally linear. They are the only finite graphs having the stronger property that every pair of vertices (adjacent or not) share exactly one common neighbor.{{r|ers}} More generally every [[Cactus graph#Triangular cactus|triangular cactus graph]], a graph in which all simple cycles are triangles and all pairs of simple cycles have at most one vertex in common, is locally linear.{{r|fp}}
Let <math>G</math> and <math>H</math> be any two locally linear graphs, select a triangle from each of them, and glue the two graphs together by identifying corresponding pairs of vertices in these triangles (this is a form of the [[clique-sum]] operation on graphs). Then the resulting graph remains locally linear.{{r|z}}
Line 159:
| volume = 23
| year = 1996}}</ref>
<ref name=fp>{{citation
| last1 = Farley | first1 = Arthur M.
| last2 = Proskurowski | first2 = Andrzej
| doi = 10.1002/net.3230120404
| issue = 4
| journal = Networks
| mr = 686540
| pages = 393–403
| title = Networks immune to isolated line failures
| volume = 12
| year = 1982}}; see in particular p. 397: "We call the resultant network a triangle cactus; it is a cactus network in which every line belongs to exactly one triangle."</ref>
<ref name=hns>{{citation
| |||