Odd mean labeling of the graphs $P_{a,b}

144 R. VASUKI AND A. NAGARAJAN
0
2 24 26 48 50
6 28 30 52 54
10 32 34 56 58
14 15 38 39 62
18 19 42 43 66
22 23 46 47 70
P 6,6
Figure 3.
Theorem 2.2. P 2r+1,2m+1 is an odd mean graph for all values of r and m.
Proof. Let v i 0, v i 1, v i 2, . . . , v i 2r+1 be the vertices of the i th copy of the path of length
2r + 1 where i = 1, 2, . . . , 2m + 1, v i 0 = u and v i 2r+1 = v for all i. We observe that the
number of vertices of the graph P 2r+1,2m+1 is 2r(2m + 1) + 2 and the number of edges
of the graph is (2r + 1)(2m + 1).
Define f on V (P 2r+1,2m+1 ) as follows:
f(u) =0,
f(v) =2(2r + 1)(2m + 1) − 1,
f(v2j+1) i =(4(2m + 1) + 3)j + 4i − 3, i = 1, 2, . . . , 2m + 1, j = 0, 1, 2, . . . , (r − 1)
⎧
(4(2m + 1) + 4) + (4(2m + 1) + 3)(j − 1)
⎪⎨ +4(i − 1),
1 ≤ i ≤ m
and f(v2j) i = (2m + 2) + (4(2m + 1) + 3)(j − 1)
+4(i − (m + 1)), m + 1 ≤ i ≤ 2m + 1,
⎪⎩
j = 1, 2, . . . , r.
It can be verified that the label of the edges of the graph are 1, 3, 5, . . . , 2q −1. Hence,
P 2r+1,2m+1 is an odd mean graph.
For example, an odd mean labeling of the graph P 7,7 is shown in Figure 4. □
1 32 29 60 57 88
5 36 33 64 61 92
9 40 37 68 65 96
13 16 41 44 69 72
❍
0 ❍
97
❍
❍
❍
❍ 17 20 45 48 73 76
❍
❍
21 24 49 52 77 80
25 28 53 56 81 84
P 7,7
Figure 4.
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