Magic Graph

An edge-magic graph is a labeled graph with  graph edges labeled with distinct elements  so that the sum of the graph edge labels at each graph vertex is the same.

A vertex-magic graph labeled graph vertices which give the same sum along every straight line segment. No magic pentagrams can be formed with the number 1, 2, ..., 10 (Trigg 1960; Langman 1962, pp. 80-83; Dongre 1971; Richards 1975; Buckley and Rubin 1977-1978; Trigg 1998), but 168 almost magic pentagrams (in which the sums are the same for four of the five lines) can. The figure above show a magic pentagram with sums 24 built using the labels 1, 2, 3, 4, 5, 6, 8, 9, 10, and 12 (Madachy 1979).

REFERENCES

Buckley, M. R. W. and Rubin, F. Solution to Problem 385. "Do Pentacles Exists?" J. Recr. Math. 10, 288-289, 1977-78.

Doob, M. "Characterization of Regular Magic Graphs." J. Comb. Th. B 25, 94-104, 1978.

Dongre, N. M. "More About Magic Star Polygons." Amer. Math. Monthly 78, 1025, 1971.Gallian, J. "Dynamic Survey of Graph Labeling." Elec. J. Combin. DS6. Dec. 21, 2018. https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.

Hartsfield, N. and Ringel, G. Pearls in Graph Theory: A Comprehensive Introduction. San Diego, CA: Academic Press, 1990.

Heinz, H. "Magic Stars." http://www.magic-squares.net/magicstar.htm.

Heinz, H. "Magic 3-D Polygons & Graphs." http://www.magic-squares.net/perimeter-3.htm.

Jezný, S. and Trenkler, M. "Characterization of Magic Graphs." Czech. Math. J. 33, 435-438, 1983.

Jeurissen, R. H. "Magic Graphs, a Characterization." Europ. J. Combin. 9, 363-368, 1988.

Langman, H. Play Mathematics. New York: Hafner, 1962.Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 98-99, 1979.

Pickover, C. A. The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions. Princeton, NJ: Princeton University Press, 2002.Richards, I. "Impossibility." Math. Mag. 48, 249-262, Nov. 1975.

Rivera, C. "Problems & Puzzles: Puzzle 013-The Prime-Magical Pentagram." http://www.primepuzzles.net/puzzles/puzz_013.htm.Silke, T. "Magic Hexagon." http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/magic-hexagon.Trigg, C. W. "Solution of Problem 113." Pi Mu Epsilon J. 3, 119-120, Fall 1960.

Trigg, C. W. "Ten Elements on a Pentagram." Eureka (Canada) 3, 5-6, Jan. 1977.

Trigg, C. W. "Almost Magic Pentagrams." J. Recr. Math. 29, 8-11, 1998.

Wynne, B. E. "Perfect Magic Icosapentacles." J. Recr. Math. 9, 241-248, 1976-77.