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graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, 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 In graph theory, an isomorphism of graphs ''G'' and ''H'' is a bijection between the vertex sets of ''G'' and ''H'' : f \colon V(G) \to V(H) such that any two vertices ''u'' and ''v'' of ''G'' are adjacent in ''G'' if and only if f(u) and f(v) ar ...
) The graphs in question may be expressed differently in terms of graph equations. Bibliography on Graph equations
/ref> 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'' = ''K''3, and ''H'' = ''K''2 are the solutions of the graph equation ''L''(''K''3) = ''T''(''K''2) and ''G'' = ''K''4, and ''H'' = ''K''3 are the solutions of the graph equation ''L''(''K''4) = ''T''(''K''3). Image: Complete graph K2.svg, :::K_2 Image: Complete graph K3.svg, :::K_3 Image: Complete graph K4.svg, :::K_4 Note that ''T''(''K''3) 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 Vasanti N. Bhat-Nayak was a mathematician whose research concerned balanced incomplete block designs, bivariegated graphs, graceful graphs, graph equations and frequency partitions. She earned a Ph.D. from the University of Mumbai The ...
, 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

{{DEFAULTSORT:Graph Equation Graph theory