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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 – ''[[Discrete Mathematics (journal)|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, [[Bhat-Nayak Vasanti N.|Vasanti N. Bhat-Nayak]], Ranjan N. Naik – ''[[Discrete Mathematics (journal)|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== | |||
{{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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