Line graph: Difference between revisions

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| url = https://research.tilburguniversity.edu/en/publications/a9a1857e-13ce-4da7-bcca-b879f9f3215f

| doi-access = free

}}. See in particular Proposition 8, p.&nbsp;262.</ref> They may also be characterized (again with the exception of {{math|''K''{{sub|8}}}}) as the [[strongly regular graph]]s with parameters {{math|srg(''n''(''n'' –− 1)/2, 2(''n'' –− 2), ''n'' –− 2, 4)}}.<ref>{{harvtxt|Harary|1972}}, Theorem 8.6, p.&nbsp;79. Harary credits this result to independent papers by L. C. Chang (1959) and [[Alan Hoffman (mathematician)|A. J. Hoffman]] (1960).</ref> The three strongly regular graphs with the same parameters and spectrum as {{math|''L''(''K''{{sub|8}})}} are the [[Chang graphs]], which may be obtained by [[Two-graph|graph switching]] from {{math|''L''(''K''{{sub|8}})}}.

The line graph of a [[bipartite graph]] is [[perfect graph|perfect]] (see [[Kőnig's theorem (graph theory)|Kőnig's theorem]]), but need not be bipartite as the example of the claw graph shows. The line graphs of bipartite graphs form one of the key building blocks of perfect graphs, used in the proof of the [[strong perfect graph theorem]].<ref>{{citation

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| doi-access = free

}}.</ref> A special case of these graphs are the [[rook's graph]]s, line graphs of [[complete bipartite graph]]s. Like the line graphs of complete graphs, they can be characterized with one exception by their numbers of vertices, numbers of edges, and number of shared neighbors for adjacent and non-adjacent points. The one exceptional case is {{math|''L''(''K''{{sub|4,4}})}}, which shares its parameters with the [[Shrikhande graph]]. When both sides of the bipartition have the same number of vertices, these graphs are again strongly regular.<ref>{{harvtxt|Harary|1972}}, Theorem 8.7, p.&nbsp;79. Harary credits this characterization of line graphs of complete bipartite graphs to Moon and Hoffman. The case of equal numbers of vertices on both sides had previously been proven by Shrikhande.</ref> It has been shown that, except for {{math|''C''{{sub|3}}}} , {{math|''C''{{sub|4}}}} , and {{math|''C''{{sub|5}}}} , all connected strongly regular graphs can be made non-strongly regular within two line graph transformations.<ref>

{{cite arxivarXiv

| eprint = 2410.16138

| title = Theoretical Insights into Line Graph Transformation on Graph Learning

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| first2 = Xingyue

| year = 2024

| class = cs.LG

}}

</ref> The extension to disconnected graphs would require that the graph is not a disjoint union of {{math|''C''{{sub|3}}}}.

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| title = Graph-theoretic concepts in computer science (Aachen, 1995)

| volume = 1017

| year = 1995}}.| isbn = 978-3-540-60618-5

}}.

*{{citation

| last1 = Evans | first1 = T.S.

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| volume = 15

| year = 1978| s2cid = 37062237

| url = http://infoscience.epfl.ch/record/88504

}}.

*{{citation