HARMONIOUS AND VERTEX GRACEFUL LABELING ON PATH AND STAR RELATED GRAPHS

In this paper, we show that B 2 (n,n) is harmonious (4) , P n i is harmonious (3), Pn × Cm is vertex graceful for n ≥ 2, and m ≥ 5,m is odd, B 2 (n,n) is vertex graceful (4) , P n


Introduction
Graph labeling, where the vertices are assigned values subject to certain conditions have often been motivated by practical problems. Labelled graphs serves as useful mathematical models for a broad range of applications such as coding url: www.acadpubl.eu theory, including the design of good radar type codes, synch-set codes, missile guidance codes and convolution codes with optimal autocorrelation properties.They facilitate the optimal non standard encoding of integers.
All graphs in this paper are finite, simple graphs with no loops or multiple edges. The symbols V (G) and E (G) denote the vertex set and edge set of the graph G. A graph with p vertices and q edges is called G(p,q) graph. Harmonious graphs naturally arose in the study by Graham and Sloane [1] of modular version of additive base problems stemming from error correction codes. They obtained some graphs are harmonious.
Definition 1.2. A graph G with p vertices and q edges is said to be vertex graceful if a labeling f : V (G) → {1, 2, 3...p} exists in such a way that the induced labeling f * : The concept of vertex graceful was introduced by Lee, Pan and Tsai in 2005. Definition 1.3. For a simple connected graph G the Square of graph G is denoted by G 2 and defined as the graph with the same vertex set as of G and two vertices are adjacent in G 2 if they are at a distance 1 or 2 apart in G.

Main Results of Harmonious Labeling on Path and Star Related
Graphs Let G be the graph B 2 (n, n), then |V (G)| = 2n + 2 and |E(G)| = 4n + 1. We define the vertex labeling f : V (G) → {0, 1, 2, 3..., (q − 1)} as follows: It is clear that vertex set labeling and edge set labeling are distinct. Hence, the B 2 (n, n) is harmonious graph ∀n.   Remark 2.3. Let α be the collection of paths P n i , where n is odd, and Theorem 2.4. The graph P n i is harmonious graph, ∀n.
Proof. Let G = P n i be a graph with p = 2n vertices and q = (2n − 1). The required vertex labeling f : V (G) → {0, 1, 2, · · · , q − 1} is as follows: It is clear that vertex set labeling and edge set labeling are distinct. Hence, the graph G is harmonious graph ∀n.

Main Results of Vertex Graceful labeling on Path and Star Related Graphs
Theorem 3.1. The graph P n × C m is a vertex graceful graph, ∀n, n ≥ 2 and m ≥ 5, m is odd.
Proof. Consider the graph G = P n ×C m with nm vertices and q = (2n−1)m edges. Suppose that the vertices v j i ; 1 ≤ i ≤ m and j = 0, 1, 2, · · · , n of the cycle C m run consecutively with v j 1 joined to v j m . The required vertex labeling f : V (G) → 1, 2, · · · , p is as follows: and j is odd 1 ≤ i ≤ n and i is odd and n + 1 2 + nj; 1 ≤ i ≤ n and i is odd, 1 ≤ j ≤ n and j is odd n + 1 + i 2 + nj 2 ; 1 ≤ j ≤ n and i is odd and j = 0, 2, 4, · · · , n Let A,B,C are denote the edge set.

and j is odd}
It is clear that vertex set labeling and edge set labeling are distinct. Hence, the P n × C m is a vertex graceful graph, for n ≥ 2 and m ≥ 5, m is odd.
It is clear that vertex set labeling and edge set labeling are distinct. Hence the graph G vertex graceful f orall n, n is odd.   Definition 3.7. The graph L n = P n × P 2 is called the ladder.
Theorem 3.8. The graph L n • K 1 is a vertex graceful ∀n.
Proof. consider the graph Let w i be pendent vertex adjacent to u i and let z i be the pendent vertex adjacent to v i . The required vertex labeling f : V (G) → {1, 2, ..., p} is as follows: It is clear that vertex set labeling and edge set labeling are distinct. Then the graph G = L n • K 1 is vertex graceful ∀n.
Theorem 3.9. The graph P n × P 2 is a vertex graceful ∀n, n is odd Proof. Consider the graph G = P n × P 2 with 2n vertices and q = 3n − 2 edges. Let v 1j and v 2j be the first and second row vertices of G respectively for 1 ≤ j ≤ n. The required vertex labeling f : V (G) → {1, 2, · · · , p} is as follows: v 1j = j + 1 2 , 1 ≤ j ≤ n; j is odd v 1j = n + j + 1 2 , 1 ≤ j ≤ n; j is even v 2j = 3n + j 2 , 1 ≤ j ≤ n;j is odd v 2j = 2n + j 2 , 1 ≤ j ≤ n; j is even Let A, B, C denote edge set.
Illustration 3.10. A vertex graceful graph P 7 × P 2 is shown in the figure 6.