Pronic Heron Mean Graphs 1R

The International journal of analytical and experimental modal analysis ISSN NO:0886-9367

Pronic Heron Mean Graphs 1R. Sophia Porchelvi [email protected] 2S. Akila Devi [email protected] 1,2 A. D. M College For Women, Nagapattinam, Tamil Nadu, India.

Abstract: In this paper we introduce a new type of labeling of graphs using pronic numbers called Pronic Heron Mean Labeling and discuss the existence and non-existence of such labeling for certain classes of graphs.

Keywords−Heron Mean Labeling, P-Heron Mean Labeling

I.Introduction 1. Preliminaries and Definitions: Throughout this paper, by a graph we mean a finite, undirected, simple and connected graph. Let G (p, q) be a graph with p vertices and q edges. For standard notations and terminology in graph theory, we follow [2]. For a detailed survey of mean labeling and Heron mean labeling, we refer to [1] and [3]. A pronic number is a number which is the product of two number of the form n (n + 1). The nth pronic number is the sum of the first n even integers. 2 is the only prime pronic number and is the only pronic number in the Fibonacci sequence. Let G (p, q) be a simple, finite, connected and undirected graph with p ≥ 2. A pronic heron mean labeling of a graph G is a bijection√f : V (G) → {0, 2, 6, 12, ..., p (p +√ 1)} such that the resulting edge labels ∗ f(u)+f(v)+ f(u)f(v) ∗ f(u)+f(v)+ f(u)f(v) obtained by f (uv) = d 3 e or f (uv) = b 3 c for every uv ∈ E(G) are all distinct. A graph G (p, q) obtained from a path Pm by attaching exactly two pendant edges to each of its

internal vertices of the path is called a twig graph and it is denoted by Tm. In general Tm has 3m−4 vertices

and 3m − 5 edges. A triangular snake TSn is obtained from the path Pn with vertices {u1, u2, u3, ..., un} by

joining a new vertex vi to uiui+1 for (0 ≤ i ≤ (n − 1)). TSn has (2n − 1) vertices and (3n − 3) edges. The

corona product of two graphs G1 and G2 is the graph formed by taking one copy of G1 and V (G1) copies of th th G2 where i vertex of G1 is adjacent to every vertex in the i copy of G2. II. Main Theorems 2. For certain connected graphs

Theorem 2.1 Path graph Pn, n ≥ 3, admits pronic heron mean labeling. Proof:

Let {v1, v2..., vn} be the vertices of Pn, n ≥ 3 and let ei = vivi+1. Define a bijection f : V (G) → ∗ {p1, ..., pn} by f(vi) = pi, i = 1, 2, ..., n. For the vertex labeling above, the induced function f : E(G) → 2 2 2 ∗ 2 {2 , 3 , ...n } is defined by f (vivi+1) = (i + 1) , i = 1, 2, ...n − 1. In the view of above defined labeling, the path graph admits pronic heron mean labeling.

Theorem 2.2 Cycle graph Cn, n ≥ 3 admits pronic graceful labeling. Proof:

Let { v,v2..., vn} be the vertices of Cn, n ≥ 3 and ei = vivi+1.

Consider Cn,for n = 4, the vertex labeling and the edge labeling are given by

f(v1) = p1 f(v2) = p2 f(v3) = p4 f(v4) = p5 ∗ ∗ ∗ ∗ f (v1v2) = 4 f (v2v3) = 12 f (v3v4) = 16 f (v1v4) = 6

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Consider Cn,for other n ∈ N.

Deﬁne a bijection f : V (G) → {p1, p2, ..., pn} by f(vi) = pi, i = 1, 2, ..., n and the induced function f ∗ : E(G) → N is deﬁned by

n2 + n + 2 + p2(n)2 + 2n f ∗(v v ) = (i + 1)2, i = 1, 2, ..., n − 1 f ∗(v v ) = d e i i+1 1 n 3 In the view of above deﬁned labeling, the cycle graph admits pronic graceful labeling.

Theorem 2.3 Complete graph Kn, n ≥ 4 does not admit pronic graceful labeling. Proof:

Let { v0, v1, v2..., vn−1} be the vertices of Kn, n ≥ 4. Since any pair of vertices of Kn are adjacent,

the edge label 9 appears twice for the pronic pairs (p2, p3) and (p1, p4). Hence the complete graph does not admit pronic graceful labeling.

Theorem 2.4 Comb graph Pn K1, n ≥ 3, admits pronic heron mean labeling. Proof: 0 0 0 Let {v1, v2..., vn} be the vertices of Pn, n ≥ 3 Let {v1, v2, ..., vn} be the pendant edges attached to the corresponding vertices of Pn.

Deﬁne a bijection f : V (G) → {p1, p2, ..., p2n} by

0 f(vi) = p2i−1, i = 1, 2, ...n f(vi) = p2i, i = 1, 2, ...n

For the vertex labeling above, the induced edge function f ∗ : E(G) → N is deﬁned by

∗ ∗ 0 2 f (vivi+1) = 2i(2i + 1), i = 1, 2, 3, ...n − 1 f (vivi) = (2i) , i = 1, 2, 3, ...n

In the view of above deﬁned labeling, the comb graph admits pronic heron mean labeling.

Theorem 2.5 Twig graph Tn, n ≥ 3, admits pronic heron mean labeling. Proof: 0 0 0 ” ” ” Let {v1, v2..., vn} be the vertices of Pn, n ≥ 3. Let {v2, v3, ..., vn−2} and {v2, v3, ..., vn−2} be the pendant edges attached to each of the corresponding internal vertices of Pn.

Deﬁne a bijection f : V (G) → {p4, p5, ..., p3n−4} by

0 f(vi) = pi, i = 1, 2 f(vi+2) = p3i+3, i = 0, 1, 2, ...n − 3 ” f(vi) = p3i−4, i = 3, 4, .., n f(vi+2) = p3i+4, i = 0, 1, 2, ...n − 3

For the above vertex labeling, the induced edge function f ∗ : E(G) → N is deﬁned by

∗ 2 ∗ 2 f (vivi+1) = (2i) , i = 1 f (vivi+1) = (3i − 2) , i = 2, 3, ...n − 1 ∗ 0 2 ∗ ” f (vivi) = (3i − 3) , i = 2, 3, ...n − 1 f (vivi ) = 3(i − 1)(3i − 2), i = 2, 3, ...n − 1

In the view of above deﬁned labeling, the twig graph admits pronic heron mean labeling.

Theorem 2.6 Triangular Snake TSn admits pronic heron mean labeling. Proof:

Let {v1, v2..., vn} be the vertices of the path Pn, n ≥ 3 and ui for 1 ≤ i ≤ n − 1 be the vertices of

TSn to which the edges uiui+1 are joining to. Deﬁne a bijection f : V (G) → {p1, p2, ..., p2n−1} by

f(vi) = p2i−1, i = 1, 2, ...n f(ui) = p2i, i = 1, 2, ...n − 1

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For the vertex labeling above, the induced edge function f ∗ : E(G) → N is deﬁned by

∗ ∗ 2 f (vivi+1) = 2i(2i + 1)i = 1, 2, 3...n − 1 f (uivi+1) = (2i + 1) , i = 1, 2, 3...n − 1 ∗ 2 f (viui) = (2i) , i = 1, 2, 3...n − 1

In the view of above deﬁned labeling, the triangular snake admits pronic heron mean labeling.

Theorem 2.7 Corona product of Cn K1 admits pronic graceful labeling. Proof: 0 0 0 Let {v1, v2..., vn} be the vertices of Cn, n ≥ 3. Let {v1, v2, ..., vn} be the pendant edges attached to the correponding vertices Cn. Deﬁne a bijection f : V (G) → {p1, p2, ..., p2n} by

0 f(vi) = p2i−1, i = 1, 2, ...n f(vi) = p2i, i = 1, 2, ...n

For the vertex labeling above, the induced edge function f ∗ : E(G) → N is deﬁned by

2 + 2n(2n − 1) + p4n(2n − 1) f ∗(v v ) = 2i(2i + 1), i = 1, 2, 3, ...n − 1 f ∗(v v ) = d e i i+1 1 n 3 ∗ 0 2 f (vivi) = (2i) , i = 1, 2, 3, ...n

In the view of above deﬁned labeling, the snake Cn K1 admits pronic heron mean labeling.

Theorem 2.8 Corona product of Cn 2K1 admits pronic graceful labeling. Proof: 0 0 0 ” ” ” Let {v1, v2..., vn} be the vertices of Cn, n ≥ 3 Let {v1, v2, ..., vn−2} and {v1, v2, ..., vn−2} be the pendant edges attached to the correponding vertices of Cn.

Deﬁne a bijection f : V (G) → {p1, p2, ..., p3n} by

0 ” f(vi) = p3i−2, i = 1, 2, ...n; f(vi) = p3i−1, i = 1, 2, ...n; f(vi ) = p3i, i = 1, 2...n.

For the vertex labeling above, the induced edge function f ∗ : E(G) → N is deﬁned by

∗ 2 ∗ 0 2 f (vivi+1) = (3i) , i = 1, 2, 3, ...n − 1 f (vivi) = (3i − 1) , i = 1, 2, 3, ...n 2 + (3n − 1)(3n − 2) + p2(3n − 1)(3n − 2) f ∗(v v ) = d e f ∗(v v”) = 3i(3i − 1), i = 1, 2, 3, ...n 1 n 3 i i

In the view of above deﬁned labeling, the snake Cn 2K1 admits pronic heron mean labeling.

Theorem 2.9 Generalized Peterson graph P (n, 1) admits pronic graceful labeling. Proof:

Let {v0, v1, v2, v3, ..., vn−1} be the inner vertices and {u0, u1, u2, u3, ..., un−1} be the outer vertices of

P (n, 1). Deifne a bijection f : V (G) → {p1, p2, ..., p2n} as follows:

f(ui) = p2i+2, i = 0, 1, 2, ...n − 1 f(vi) = p2i+1, i = 0, 1, 2, ...n − 1

For the vertex labeling above, the induced edge function f ∗ : E(G) → N is deﬁned below and hence the edge labels are distinct. 2 + (2n + 1)(2n + 2) + p2(2n + 1)(2n + 2) f ∗(v v ) = (2i + 2)(2i + 3), i = 0, 1, 2, ...n − 1; f ∗(v v ) = d e i i+1 0 n 3 6 + (2n + 2)(2n + 3) + p6(2n + 2)(2n + 3) f ∗(u u ) = (2i + 3)(2i + 4), i = 0, 1, 2, ...n − 1; f ∗(v v ) = d e i i+1 0 n 3 ∗ 2 f (uivi) = (2i + 2) , i = 0, 1, 2, ...n − 1

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3. For some disconnected graphs:

Theorem 3.1 Pm ∪ Pn, m, n ≥ 3 admits pronic graceful labeling. Proof:

Let {v1, v2..., vm} be the vertices of Pn, n ≥ 3 with ei = vivi+1 and {u1, u2..., un} be the vertices of

Pn, n ≥ 3 with ei = uiui+1. Deﬁne a bijection f : V (G) → {p1, ..., pm+n} by

f(vi) = pi, i = 1, 2, ..., m f(ui) = pm+i, i = 1, 2, ..., n

For the vertex labeling above, the induced function f ∗ : E(G) → N is deﬁned by

∗ 2 ∗ 2 f (vivi+1) = (i + 1) , i = 1, 2, ...n − 1 f (uiui+1) = (m + i + 1) , i = 1, 2, ...n − 1

In the view of above deﬁned labeling, the graph Pm ∪ Pn, m, n ≥ 3 admits pronic heron mean labeling.

Theorem 3.2 (Pn K1) ∪ Pm, m, n ≥ 3 admits pronic graceful labeling. Proof: 0 0 0 Let {v1, v2..., vn} be the vertices of Pn, n ≥ 3 Let {v1, v2, ..., vn} be the pendant edges attached to the corresponding vertices of Pn and {u1, u2..., um} be the vertices of Pm, m ≥ 3 with ei = uiui+1.

Deﬁne a bijection f : V (G) → {p1, p2, ..., p2n+m} by

0 f(vi) = p2i−1, i = 1, 2, ...n; f(ui) = p2n+i, i = 1, 2, ..., m; f(vi) = p2i, i = 1, 2, ...n

For the vertex labeling above, the induced edge function f ∗ : E(G) → N is deﬁned by

∗ ∗ 2 f (vivi+1) = 2i(2i + 1), i = 1, 2, 3, ...n − 1 f (uiui+1) = (2n + i + 1) , i = 1, 2, ...m − 1. ∗ 0 2 f (vivi) = (2i) , i = 1, 2, 3, ...n.

In the view of above deﬁned labeling, the graph (Pn K1)∪Pm, m, n ≥ 3 admits pronic heron mean labeling.

Theorem 3.3 Cm ∪ Pn, m ≥ 5, n ≥ 3 admits pronic heron mean labeling. Proof:

Let { u1, u2..., un} be the vertices of Cm, n ≥ 5 with ei = uiui+1 and Let {v1, v2..., vm} be the

vertices of Pm, m ≥ 3 with ei = vivi+1. Deﬁne a bijection f : V (G) → {p1, ..., pn+m} by

f(ui) = pi, i = 1, 2, ..., n f(vi) = pn+i, i = 1, 2, ..., m

For the vertex labeling above, the induced function f ∗ : E(G) → N is deﬁned by

n2 + n + 2 + p2(n)2 + 2n f ∗(u u ) = (i + 1)2, i = 1, 2, ..., n − 1 f ∗(u u ) = d e i i+1 1 n 3 ∗ 2 f (vivi+1) = (n + i + 1) , i = 1, 2, ...m.

In the view of above deﬁned labeling, the graph Cm ∪ Pn, m ≥ 5, n ≥ 3 admits pronic heron mean labeling.

Theorem 3.4 mK3, m ≥ 2 admits pronic heron mean labeling. Proof: i i i Let the vertex set of mK3 be V = {V1 ∪ V2 ∪ ... ∪ Vm} where Vi = {v1, v2, v3}. Deﬁne a bijection f : V (mK3) → {p1, p2, ..., p3m} by

i i i f(v1) = p3i−2, i = 1, 2, ...m f(v2) = p3i−1, i = 1, 2, ...m f(v3) = p3i, i = 1, 2, ...m

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For the vertex labeling above, the induced edge function f ∗ : E(G) → N is deﬁned by

∗ i 2 f (v1v2) = (3i − 1) , i = 1, 2, 3, ...m ∗ i 2 f (v2v3) = (3i) , i = 1, 2, 3, ...m ∗ i 2 f (v1v3) = 3(3i − i), i = 1, 2, 3, ...m

In the view of above deﬁned labeling, the graph mK3 admits pronic heron mean labeling.

Theorem 3.5 Cn ∪ mK3, n ≥ 5, m ≥ 2 admits pronic heron mean labeling. Proof:

Let { u1, u2..., un} be the vertices of Cn, n ≥ 5 with ei = uiui+1 and let V = {V1 ∪ V2 ∪ ... ∪ Vm} i i i where Vi = {v1, v2, v3} be the vertices of mK3 . Deﬁne a bijection f : V (G) → {p1, p2, ..., pn+m} by

i f(ui) = pi, i = 1, 2, ..., n f(v1) = pn+3i−2, i = 1, 2, ...m i i f(v2) = pn+3i−1, i = 1, 2, ...m f(v3) = pn+3i, i = 1, 2, ...m

For the above vertex labeling, the induced function f ∗ : E(G) → N is deﬁned by

∗ 2 ∗ i 2 f (uiui+1) = (i + 1) , i = 1, 2, ..., n − 1 f (v1v2) = (n + 3i − 1) , i = 1, 2, 3, ...m n2 + n + 2 + p2(n)2 + 2n f ∗(u u ) = d e f ∗(v v )i = (n + 3i)2, i = 1, 2, 3, ...m 1 n 3 2 3 ∗ i f (v1v3) = (n + (3i − 1))(n + 3i), i = 1, 2, 3, ...m

In the view of above deﬁned labeling, the graph Cn ∪ mK3 admits pronic heron mean labeling.

Theorem 3.6 (Cn K1) ∪ Pm, m, n ≥ 3 admits pronic graceful labeling. Proof: 0 0 0 Let {v1, v2..., vn} be the vertices of Cn, n ≥ 3 Let {v1, v2, ..., vn} be the pendant edges attached to the corresponding vertices of Cn and let {u1, u2..., um} be the vertices of Pn, n ≥ 3 with ei = uiui+1.

Deﬁne a bijection f : V (G) → {p1, p2, ..., p2n+m} by

0 f(vi) = p2i−1, i = 1, 2, ...n f(vi) = p2i, i = 1, 2, ...n f(ui) = p2n+i, i = 1, 2, ..., m

For the vertex labeling above, the induced edge function f ∗ : E(G) → N is deﬁned by

∗ ∗ 2 f (vivi+1) = 2i(2i + 1), i = 1, 2, 3, ...n − 1 f (uiui+1) = (2n + i + 1) , i = 0, 1, ..m ∗ 0 2 f (vivi) = (2i) , i = 1, 2, 3, ...n 2 + 2n(2n − 1) + p4n(2n − 1) f ∗(v v ) = d e 1 n 3

In the view of above deﬁned labeling, the corona product Cn K1 graph admits pronic heron mean labeling.

Theorem 3.7 (Cn 2K1) ∪ Pm, m, n ≥ 3 admits pronic heron mean labeling. Proof: 0 0 0 ” ” ” Let {v1, v2..., vn} be the vertices of Cn, n ≥ 3. Let {v1, v2, ..., vn−2} and {v1, v2, ..., vn−2} be the pendant edges attached to the corresponding vertices of Cn and let {u1, u2..., um} be the vertices of Pm,

n ≥ 3 with ei = uiui+1.

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Deﬁne a bijection f : V (G) → {p1, p2, ..., p3n+m} by

f(vi) = p3i−2, i = 1, 2, ...n f(ui) = p3n+i, i = 1, 2, ..., m 0 f(vi) = p3i−1, i = 1, 2, ...n ” f(vi ) = p3i, i = 3

For the vertex labeling above, the induced edge function f ∗ : E(G) → N is deﬁned by

∗ 2 ∗ 2 f (vivi+1) = (3i) , i = 1, 2, 3, ...n − 1 f (uiui+1) = (3n + i + 1) , i = 0, 1, ..m 2 + (3n − 1)(3n − 2) + p2(3n − 1)(3n − 2) f ∗(v v ) = d e 1 n 3 ∗ 0 2 ∗ ” f (vivi) = (3i − 1) , i = 1, 2, 3, ...n f (vivi ) = 3i(3i − 1), i = 1, 2, 3, ...n

In the view of above deﬁned labeling, the Corona product of Cn 2K1 admits pronic heron mean labeling.

III. Conclusion

This paper exhibits the concept of pronic heron mean labeling and investigated the existence and non-existence of such labeling for certain classes of graphs. The authors are highly thankful to the anony- mous referees for their valuable suggestions.

IV. References 1. Joseph A. Gallion, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics, Vol. no. 19, 2012. 2. J. Gross and J. Yellen, Graph Theory and its Applications, CRC Press, 2004. 3. Sambath Kumar. R, Narasimhan. G, Nagaraja. K. M, Heron Mean Labeling of Graphs, International Journal of Recent Scientiﬁc Research, Vol 8, Sep 2017, pp(19808-19811)

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