A magic graph is a

graph 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 ...

whose edges are labelled by the first ''q'' positive

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s, where ''q'' is the number of edges, so that the sum over the edges incident with any vertex is the same, independent of the choice of vertex; or it is a graph that has such a labelling. The name "magic" sometimes means that the integers are any positive integers; then the graph and the labelling using the first ''q'' positive integers are called supermagic. A graph is vertex-magic if its vertices can be labelled so that the sum on any edge is the same. It is total magic if its edges and vertices can be labelled so that the vertex label plus the sum of labels on edges incident with that vertex is a constant. There are a great many variations on the concept of magic labelling of a graph. There is much variation in terminology as well. The definitions here are perhaps the most common. Comprehensive references for magic labellings and magic graphs are Gallian (1998), Wallis (2001), and Marr and Wallis (2013).

Magic squares

A semimagic square is an ''n'' × ''n'' square with the numbers 1 to ''n''² in its cells, in which the sum of each row and column is the same. A semimagic square is equivalent to a magic labelling of the

complete bipartite graph In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory i ...

''K''_''n'',''n''. The two vertex sets of ''K''_''n'',''n'' correspond to the rows and the columns of the square, respectively, and the label on an edge ''r''_''i'' ''s''_''j'' is the value in row ''i'', column ''j'' of the semimagic square. The definition of semimagic squares differs from the definition of

magic square In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...

s in the treatment of the diagonals of the square. Magic squares are required to have diagonals with the same sum as the row and column sums, but for semimagic squares this is not required. Thus, every magic square is semimagic, but not vice versa.

References

* Nora Hartsfield and

Gerhard Ringel Gerhard Ringel (October 28, 1919 in Kollnbrunn, Austria – June 24, 2008 in Santa Cruz, California) was a German mathematician. He was one of the pioneers in graph theory and contributed significantly to the proof of the Heawood conjecture (now ...

(1994, 2003), ''Pearls in Graph Theory'', revised edition. Dover Publications, Mineola, N.Y. Section 6.1. * W. D. Wallis (2001), ''Magic Graphs''. Birkhäuser Boston, Boston, Mass. * Alison M. Marr and W. D. Wallis (2013), ''Magic Graphs''. Second edition. Birkhäuser/Springer, New York. ; 978-0-8176-8391-7 * Joseph A. Gallian (1998)
A dynamic survey of graph labeling.
''Electronic Journal of Combinatorics'', vol. 5, Dynamic Survey 6. Updated many times. Graph theory Magic shapes {{graph-stub