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 bijectionpf : V (G) ! f0; 2; 6; 12; :::; p (p +p 1)g 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 2 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 fu1; u2; u3; :::; ung 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 fv1; v2:::; vng be the vertices of Pn, n ≥ 3 and let ei = vivi+1. Define a bijection f : V (G) ! ∗ fp1; :::; png by f(vi) = pi; i = 1; 2; :::; n. For the vertex labeling above, the induced function f : E(G) ! 2 2 2 ∗ 2 f2 ; 3 ; :::n g 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 f v;v2:::; vng 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 Volume XII, Issue III, March/2020 1 Page No:2207 The International journal of analytical and experimental modal analysis ISSN NO:0886-9367 Consider Cn,for other n 2 N: Define a bijection f : V (G) ! fp1; p2; :::; png 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 f v0; v1; v2:::; vn−1g 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 fv1; v2:::; vng be the vertices of Pn, n ≥ 3 Let fv1; v2; :::; vng be the pendant edges attached to the corresponding vertices of Pn. Define a bijection f : V (G) ! fp1; p2; :::; p2ng 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 fv1; v2:::; vng be the vertices of Pn, n ≥ 3: Let fv2; v3; :::; vn−2g and fv2; v3; :::; vn−2g be the pendant edges attached to each of the corresponding internal vertices of Pn. Define a bijection f : V (G) ! fp4; p5; :::; p3n−4g 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 fv1; v2:::; vng 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) ! fp1; p2; :::; p2n−1g by f(vi) = p2i−1; i = 1; 2; :::n f(ui) = p2i; i = 1; 2; :::n − 1 Volume XII, Issue III, March/2020 2 Page No:2208 The International journal of analytical and experimental modal analysis ISSN NO:0886-9367 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 fv1; v2:::; vng be the vertices of Cn, n ≥ 3: Let fv1; v2; :::; vng be the pendant edges attached to the correponding vertices Cn. Define a bijection f : V (G) ! fp1; p2; :::; p2ng 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 fv1; v2:::; vng be the vertices of Cn, n ≥ 3 Let fv1; v2; :::; vn−2g and fv1; v2; :::; vn−2g be the pendant edges attached to the correponding vertices of Cn. Define a bijection f : V (G) ! fp1; p2; :::; p3ng 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 fv0; v1; v2; v3; :::; vn−1g be the inner vertices and fu0; u1; u2; u3; :::; un−1g be the outer vertices of P (n; 1). Deifne a bijection f : V (G) ! fp1; p2; :::; p2ng 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 Volume XII, Issue III, March/2020 3 Page No:2209 The International journal of analytical and experimental modal analysis ISSN NO:0886-9367 3.