SOME NEW SUM PERFECT SQUARE GRAPHS

A (p, q) graph G = (V,E) is called sum perfect square if for a bijection f : V (G) → {0, 1, 2, . . . , p − 1} there exists an injection f∗ : E(G) → N defined by f∗(uv) = (f(u)) + (f(v)) + 2f(u) · f(v), ∀uv ∈ E(G). Here f is called sum perfect square labeling of G. In this paper we derive several new sum perfect square graphs. AMS Subject Classification: 05C78.


Introduction
We consider simple, finite, undirected graph G = (p, q) (with p vertices and q edges).The vertex set and the edge set of G are denoted by V (G) and E(G) respectively.For all other terminology and notations we follow Harary [1].
Sonchhatra and Ghodasara [4] initiated the study of sum perfect square graphs.Due to [4] it becomes possible to construct a graph, whose all edges can be labeled by different perfect square integers.In [4] the authors proved that P n , C n , C n with one chord, C n with twin chords, tree, K 1,n , T m,n are sum perfect square graphs.
In this paper we prove that half wheel, corona, middle graph, total graph, K 1,n + K 1 , K 2 + mK 1 are sum perfect square graphs.Definition 1.1.Let G = (p, q) be a graph.A bijection f : V (G) → {0, 1, 2, . . ., p − 1} is called sum perfect square labeling of G, if the induced function f * : E(G) → N defined by f * (uv) = (f (u) A graph which admits sum perfect square labeling is called sum perfect square graph.(2) x, y ∈ E(G) are adjacent.
(3) x ∈ V (G), y ∈ E(G) and y is incident to x. Definition 1.5.Half wheel graph, denoted by HW n is constructed by the following steps.
Step 1: Consider a star K 1,n .Let {v 1 , v 2 , . . ., v n } be the pendant vertices of K 1,n and v be the apex vertex of K 1,n .
Step 2: Add an edge between v i and v i+1 , 1

Main Results
Observation 1: If G = (V, E) is not sum perfect square graph, then its supergraph is also not sum perfect square graph, but the converse may not be true.In [4] Sonchhatra and Ghodasara posed the following conjecture.Here we prove this conjecture by using the principle of mathematical induction on number of vertices of the graph.Theorem 2.2.An odd simple graph G with δ(G) = 3 is not sum perfect square.
Proof.For any graph G = (V, E) with |V (G)| = n, since d(v) ≥ 3, ∀v ∈ G, n must be even and n ≥ 4. We prove this conjecture by using principle of mathematical induction on number of vertices n of graph G.
Step 2. Suppose the result is true for n = k, 4 < k < n.
Step 3. Let G ′ = (V ′ , E ′ ) be the graph with {v} is the graph with k vertices, where v is any arbitrary vertex of G ′ .By induction hypothesis, H is not a sum perfect square graph.Since G ′ is a supergraph of H, it is not a sum perfect square graph (Observation 1).
We define a bijection f : Injectivity for edge labels: n ) = 4 are three smallest edge labels among all the edge labels in this graph.
(2) {f * (e ) is larger than the largest edge label of {f * (e i ), 1 ≤ i ≤ ⌊ n 2 ⌋} and smaller than the smallest edge label of {f * (e Hence we only need to prove the following.
(1) {f * (e (2) {f * (e i ), Assume if possible {f * (e i ), 2 , which contradicts with the choice of i and n as i, n ∈ N. Assume if possible {f * (e 4 , which contradicts with the choice of i as i ∈ N.
The below illustration provides better idea about the above defined labeling pattern.
We define the bijection f : Let f * : E(M (P n )) → N be the induced edge labeling function defined by Injectivity for edge labels: i ) and f * (e i ), for some i, The below illustration provides the better idea of the above defined labeling pattern.
We define a bijection f : Injectivity for edge labels: 3) ) are even and f * (e i (j) ) are odd, for 1 4 , which contradicts with the choice of i, as i ∈ N. Assume if possible {f * (e 2 , which contradicts with the choice of i, as i ∈ N.So f * : E(T (P n )) → N is injective.Hence T (P n ) is sum perfect square graph, ∀n ∈ N.
The below illustration provides the better idea of the above defined labeling pattern.Theorem 2.6.HW n is sum perfect square graph, ∀n ∈ N.
Let f * : E(HW n ) → N be the induced edge labeling function defined by Injectivity for edge labels: i ) is also increasing.The largest edge label of f * (e  Theorem 2.7.K 1,n + K 1 is sum perfect square graph, ∀n ∈ N. Proof.Let V (K 1,n +K 1 ) = {v}∪{v i ; 1 ≤ i ≤ n}∪{w}, where {v 1 , v 2 , . . ., v n } are the pendant vertices and v is the apex vertex of K 1,n and w is the apex vertex corresponding to K 1 .Here We define a bijection f :

Injectivity for edge labels
The largest edge label of f * (e i ) is smaller than the smallest edge label of f * (e i 2) ).
If {f * (e i ), 1 ≤ i ≤ n} = {f * (e)} for some i, then i = n + 1 or i = −n − 1, which contradicts with the choice of i, as i ∈ N.

Definition 1 . 2 .Definition 1 . 4 .
The corona product G ⊙ H of two graphs G and H is obtained by taking one copy of G and |V (G)| copies of H and by joining each vertex of the i th copy of H to the i th vertex of G by an edge, 1 ≤ i ≤ |V (G)|.Definition 1.3.The middle graph of a graph G denoted by M (G) is the graph with vertex set V (G) ∪ E(G), where two vertices are adjacent if and only if either they are adjacent edges of G or one is a vertex of G and other is an edge incident with it.The total graph of a graph G denoted by T (G) is the graph with vertex set is V (G) ∪ E(G) where two vertices are adjacent if and only if (1) x, y ∈ V (G) are adjacent.

Figure 1 :
Figure 1: A non sum perfect square graph K 4 with sum perfect square subgraph K 4 − {e}.

Conjecture 2 . 1 .
An odd simple graph G with δ(G) = 3 is not sum perfect square.

( 1 )
i ) is smaller than the smallest edge label of f * (e i ), therefore {f * (e i ); 1≤ i ≤ n} = {f * (e (1) i ); 1 ≤ i ≤ ⌊ n 2 ⌋}.Hence the induced edge labeling f * : E(H(W n )) → N is injective.So HW n is sum perfect square graph, ∀n ∈ N.The below illustration gives the better understanding of above defined labeling pattern.