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In [[graph theory]],''' Graph equations''' are [[equations]] in which the unknowns are [[Graph theory|graphs]]. One of the central questions of graph theory concerns the notion of [[Graph isomorphism|isomorphism]]. We ask: When are two graphs the same (i.e, [[graph isomorphism]])? The graphs in question may be expressed differently in terms of graph equations.<ref>[http://www3.interscience.wiley.com/journal/113386917/abstract?CRETRY=1&SRETRY=0  Bibliography on Graph equations ]</ref>
 
What are the graphs ([[root of a function|solutions]]) ''G'' and ''H'' such that the [[line graph]] of ''G'' is same as the [[total graph]] of ''H''? (What are ''G'' and ''H'' such that  ''L''(''G'') = '' T''( ''H'') ? ).
 
For example, ''G'' = ''K''<sub>3</sub>, and ''H'' = ''K''<sub>2</sub> are the solutions of the graph equation  ''L''(''K''<sub>3</sub>)  =  ''T''(''K''<sub>2</sub>) and ''G'' = ''K''<sub>4</sub>, and ''H'' = ''K''<sub>3</sub> are the solutions of the graph equation ''L''(''K''<sub>4</sub>) = ''T''(''K''<sub>3</sub>).
 
<gallery>
 
Image: Complete graph K2.svg|<math>K_2</math>
Image: Complete graph K3.svg|<math>K_3</math>
Image: Complete graph K4.svg|<math>K_4</math>
</gallery>
 
Note that ''T''(''K''<sub>3</sub>) is a 4-[[regular graph]] on 6 vertices.
 
==Selected publications==
*  Graph equations for line graphs and total graphs, DM Cvetkovic, SK Simic &ndash; ''[[Discrete Mathematics (journal)|Discrete Mathematics]]'', 1975
* Graph equations, graph inequalities and a fixed point theorem, DM Cvetkovic, IB Lackovic, SK Simic &ndash; Publ. Inst. Math.(Belgrade)., 1976 &ndash; elib.mi.sanu.ac.yu , PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE Nouvelle série, tome 20 (34), 1976,
* Graphs whose complement and line graph are isomorphic, M Aigner &ndash; ''[[Journal of Combinatorial Theory]]'', 1969
* Solutions of some further graph equations, [[Bhat-Nayak Vasanti N.|Vasanti N. Bhat-Nayak]], Ranjan N. Naik &ndash; ''[[Discrete Mathematics (journal)|Discrete Mathematics]]'', 47 (1983) 169&ndash;175
* More Results on the Graph Equation G2= G, M Capobianco, SR Kim &ndash; Graph Theory, Combinatorics, and Algorithms: Proceedings of …, 1995 &ndash; Wiley-Interscience
* Graph equation Ln (G)= G, S Simic - Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz, 1975
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Graph Equation}}
[[Category:Graph theory]]

Revision as of 08:33, 17 January 2013

In graph theory, Graph equations are equations in which the unknowns are graphs. One of the central questions of graph theory concerns the notion of isomorphism. We ask: When are two graphs the same (i.e, graph isomorphism)? The graphs in question may be expressed differently in terms of graph equations.[1]

What are the graphs (solutions) G and H such that the line graph of G is same as the total graph of H? (What are G and H such that L(G) = T( H) ? ).

For example, G = K3, and H = K2 are the solutions of the graph equation L(K3) = T(K2) and G = K4, and H = K3 are the solutions of the graph equation L(K4) = T(K3).

Note that T(K3) is a 4-regular graph on 6 vertices.

Selected publications

  • Graph equations for line graphs and total graphs, DM Cvetkovic, SK Simic – Discrete Mathematics, 1975
  • Graph equations, graph inequalities and a fixed point theorem, DM Cvetkovic, IB Lackovic, SK Simic – Publ. Inst. Math.(Belgrade)., 1976 – elib.mi.sanu.ac.yu , PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE Nouvelle série, tome 20 (34), 1976,
  • Graphs whose complement and line graph are isomorphic, M Aigner – Journal of Combinatorial Theory, 1969
  • Solutions of some further graph equations, Vasanti N. Bhat-Nayak, Ranjan N. Naik – Discrete Mathematics, 47 (1983) 169–175
  • More Results on the Graph Equation G2= G, M Capobianco, SR Kim – Graph Theory, Combinatorics, and Algorithms: Proceedings of …, 1995 – Wiley-Interscience
  • Graph equation Ln (G)= G, S Simic - Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz, 1975

References

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  1. ↑ Bibliography on Graph equations
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