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 conne ...

, Graph equations are

equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...

s in which the unknowns are

graphs Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

. One of the central questions of graph theory concerns the notion of

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...

. 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. 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 Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...

'', 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 The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applicat ...

'', 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 in 1970 w ...

, Ranjan N. Naik – ''

'', 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