Simply Sequentially Additive Labeling of Some Special Trees

Simply sequentially additive labeling 6507
We define f on V ∪ E as follows:
f(v1) = 1, f(u0) = n+3, f(v2) = n+5,
n+
1
⎧2i
; 1≤
i ≤ 2
⎪
n+
3
⎪
n + 6 ; i = 2
n+
5
f(v1,i
) = ⎨2n
+ 9 ; i = 2
⎪
n+
7
2n + 11;
i = 2
⎪
n+
9
⎪⎩
2(n + i + 2) ; 2 ≤ i ≤ n
f(v2,j) = n+j+7 ; 1 ≤ j ≤ n
n+
1
⎧2i
+ 1`;
1 ≤ i ≤ 2
⎪
n+
3
⎪
n + 7 ; i = 2
n+
5
f(v1u0) = n+ 4, f(v2u0) = 2n + 8, f(v1
v1,
i ) = ⎨2n
+ 10 ; i = 2
⎪
n+
7
2n + 12 ; i = 2
⎪
n+
9
⎪⎩
2(n + i) + 5 ; 2 ≤ i ≤ n
f(v2,v2,j) = 2n+12 +j ; 1 ≤ j ≤ n.
(ii). If n =1, then is a Path P5. By Theorem 1.3[1], it is an SSA-graph.
Subcase 2.3: n ≡ 2 (mod 3).
Let V and E be defined as in case 1.
Define f on V ∪ E as follows:
n+
1
⎧2i
; 1 ≤ i ≤ 2
⎪
n+
3
⎪2(n
+ 3) ; i = 2
f(v1) = 1, f(v2) = n+5, f(u0) = n+3, f(v1,
i ) = ⎨
n+
5
⎪2n
+ 9 ; i = 2
⎪
n+
7
⎩2(n
+ 2 + i) ; 2 ≤ i ≤ n
f(v2,j) = n+5+j ; 1 ≤ j ≤ n.
f(v1u0) = n+ 4, f(v2u0) = 2n + 8
n+
1
⎧2i
+ 1;
1 ≤ i ≤ 2
⎪
n+
3
⎪2n
+ 7 ; i = 2
f(v1
v1,
i ) = ⎨
n+
5
⎪2n
+ 10 ; i = 2
⎪
n+
7
⎩2(n
+ i) + 5 ; 2 ≤ i ≤ n
f(v2v2,j) = 2n+10 +j ; 1 ≤ j ≤ n.
In all the above cases, the labels are distinct and satisfy the condition f(uv) = f(u)
+ f(v) for each uv ∈ E. Hence admits 1-sequentially additive labeling.
In Fig. 4, we display 1-sequentially additive labeling for .