An edge-magic graph is a labeled graph with graph edges labeled with distinct elements 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).
See also Antimagic Graph ,
Labeled Graph ,
Magic Circles ,
Magic Constant ,
Magic Cube ,
Magic Hexagon ,
Magic Square Explore with Wolfram|Alpha 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 . Oct. 30, 2025. https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6 . Hartsfield, N. and Ringel, G. Pearls in Graph Theory: A Comprehensive Introduction. 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. Madachy, J. S. Madachy's Mathematical Recreations. Pickover, C. A. The Zen of Magic Squares, Circles, and Stars: An Exhibition of Surprising Structures Across Dimensions. 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. Referenced on Wolfram|Alpha Magic Graph Cite this as: Weisstein, Eric W. "Magic Graph." From MathWorld https://mathworld.wolfram.com/MagicGraph.html
Subject classifications 
graph edges labeled with distinct elements 
so that the sum of the graph edge labels at each graph vertex is the same. 
