Further Results on Pair Sum Labeling of Trees

Let G be a (p,q) graph. An injective map f:V(G)→﹛±1,±2,...±p﹜is called a pair sum labeling if the induced edge function, fe:E(G)→Z-﹛0﹜defined by fe(uv)=f(u)+f(v) is one-one and fe(E(G)) is either of the form {±k1,±k2,...,±kq/2} or {±k1,±k2,...,±kq-1/2}∪{kq+1/2} according as q is even or odd. A graph with a pair sum labeling is called a pair sum graph. In this paper we investigate the pair sum labeling behavior of some trees which are derived from stars and bistars. Finally, we show that all trees of order nine are pair sum graphs.

is called a pair sum labeling if the induced edge function, is either of the form  

Introduction Notation 2.1:
We denote the vertex and edge sets of star 1,n K as follows: The graphs in this paper are finite, undirected and simple.and will denote the vertex set and edge set of a graph G.The cardinality of the vertex set of a graph G is called the order of G and is denoted by p.The cardinality of its edge set is called the size of G and is denoted by q.The concept of pair sum labeling has been introduced in [1].The Pair sum labeling behavior of some standard graphs like complete graph, cycle, path, bistar, and some more standard graphs are investigated in [1][2][3].Terms not defined here are used in the sense of Harary [4].All the trees of order ≤8 are pair sum have been proved in [5].Here we proved that all trees of order nine are pair sum.Let x be any real number.Then x     stands for the largest integer less than or equal to x and x     stands for the smallest inter greater than or equal to x.

Pair Sum Labeling
Theorem 2.3 [5]: All graphs of order ≤8 are pair sum.Now we derive some pair sum trees which are used for the final section.

Pair Sum Labeling of Star Related Graphs
Here we prove that some trees which are obtained from stars are pair sum.Theorem 3.1: The trees with vertex set and edge set given below are pair sum.
, (1 5) according as q is even or odd.A graph with a pair sum labeling defined on it is called a pair sum graph.
Then is a pair sum graph. 5 Proof 1): Then is a pair sum tree.
Then is a pair sum graph.
Then is a pair sum graph.

Bistar Related Graphs
In this section we show that some trees which are obtained from bistar are pair sum.Theorem 4.1: Let G be the tree with

E G E B vw w w w w vw w w w w
.
Then G is a pair sum tree.
Proof: Define a function . m n  Assign the label to as in case 1).Define Then G is a pair sum graph , , , , , Then G is a pair sum tree.and Label the vertices and as in case 1) for as in case 1).Define and Assign the label to as in case 1).Define as in case 1).Define and Then G is a pair sum graph.□ Theorem 4.5: Let G be the tree with Then G is a pair sum graph.□ Theorem 4.4: The tree G with vertex set and

E G E B w w w u uw w v vw w w uv
Then G is a pair sum tree.Proof: Define a function .
Then G is a pair sum graph.
, , , , Then G is a pair sum tree.Proof: Define a function as in case 1).Define


Case 3): .m n  Assign the label to as in case 1).Define Then G is a pair sum graph.□ Theorem 4.7: Let G be the tree with Then G is a pair sum tree.
Proof: Define a function Case 3): Then G is a pair sum graph.□ Theorem 4.8: The tree G with Then G is a pair sum graph.
Proof: Define a map .
Then G is a pair sum graph.□ Theorem 4.9: The tree G with E G E B w w w u vw w w w w   .
Then G is a pair sum graph.Proof: Define a map Case 3): .m n  Assign the label to as in case 1).Define .
Then G is a pair sum graph.□ Theorem 4.10: The tree G with E G E B w w w u vw w w vw w w   6 .Then G is a pair sum graph.
Proof: Define a map 4, Case 2): m n  Assign the label to as in case 1).Define .


Assign the label to as in case 1).Define Then G is a pair sum graph.□ Illustration 6: A pair sum labeling of the tree in theorem 4.10 with 10 m  ,  6 Case 3): .m n  Assign the label to as in case 1).Define Then G is a pair sum graph.
Proof: Define a map Then G is a pair sum graph.□ Theorem 4.12: The tree G with Case 1): .


Then G is a pair sum graph.Case 2): m n  Proof: Define a map Assign the label to as in case 1).Define .
Then G is a pair sum graph.□ Theorem 4.13: The tree G with Then G is a pair sum graph.Proof: Define a map


Case 3): .m n  Assign the label to as in case 1).Define  , 1 Then G is a pair sum graph.□ Theorem 4.14: The tree G with       , :1 3 Then G is a pair sum graph.□ Illustration 7: A pair sum labeling of the tree in theorem 4.14 with 9 m  , 6 n  is given below:

Trees of Order 9
Here we prove that all trees of order ≤9 are pair sum.
Theorem 5.1: The trees given below are pair sum.

Notation 2 . 2 :
We denote the vertex and edge sets of bistar as

Definition 2 . 1 :
Let G be a   one-one and e 

Theorem 4 . 3 :
Let G be the tree with and edge set [5]phs in case 1) to case 5) are pair sum by theorem 3.1.andcase6)to case 19) graphs are pair sum by theorem 4.1 to 4.14.□Remark5.2:The remaining trees of order 9 are pair sum by theorems in[5].Theorem 5.3: All trees of order 9 are pair sum.