By Masatsugu Tsuji
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Extra resources for Potential Theory in Modern Function Theory
Cj = n − 61 write 48 n+5 n+5 + 10 + 8j, n − 1 − 24j − 20, + 4 + 8j, 3 3 n+5 n − 1 − 24j − 16, + 8 + 8j, n − 1 − 24j − 12, 3 n+5 n+5 + 6 + 8j, n − 1 − 24j − 8, − 1 + 8j, 3 3 n+5 n − 1 − 24j − 4, + 3 + 8j, n − 1 − 24j . 6 Wheels 47 Then the desired sequence is SS6 S9 . . S n−19 S n−10 T C n−61 . . C2 C1 D. ) 2. n ≡ 37 (mod 48), n ≥ 37. n−37 Write D = ( n+5 3 + 4), and for each j = 1, 2, . . , 48 define Cj = n+5 n+5 + 6 + 8j, n − 1 − 24j − 8, + 8j, 3 3 n+5 n − 1 − 24j − 4, + 4 + 8j, n − 1 − 24j, 3 n+5 n+5 + 2 + 8j, n − 1 − 24j + 4, − 5 + 8j, 3 3 n+5 n − 1 − 24j + 8, − 1 + 8j, n − 1 − 24j + 12 .
Avadayappan et al.  give the following simple construction for a super edgemagic labeling of a path. Say n = 2m or 2m+1. Label the vertices x1 , x2 , . . , x2m , preceded by x0 if n is odd. Vertices x1 , x3 , x5 , . . , x2n−1 receive labels 1, 2, 3, . . , n, while even vertices (x0 , x2 , x4 , . . , x2n when n is odd, x2 , x4 , x6 , . . , x2n in the even case) receive n+1, n+2, . . in order. When this labeling is completed in the obvious way, the magic sum is 12 (5n + 1) , the theoretical minimum.
14 1 Preliminaries λ(x)+λ(y) cannot equal λ(z)+λ(t). Thus, λ(x)−λ(z) = λ(t)−λ(y). Therefore the n2 differences between the labels of Kn are all different. If λ is any labeling of Kn , a ruler model of λ is constructed as follows. For each vertex of Kn , place a mark distance λ(x) from the start of the ruler. The ruler can be used to measure all distances corresponding to the distance between two marks. Ruler models are discussed, for example, in [14, 15]. The ruler models derived from edge-magic injections have the following special property.
Potential Theory in Modern Function Theory by Masatsugu Tsuji