Malaya Journal of Matematik, Vol. 9, No. 1, 262-266, 2021 https://doi.org/10.26637/MJM0901/0044 Pronic Heron Mean labeling on special cases of generalized Peterson graph P(n;k) and disconnected graphs S. Akila Devi1*, R. Sophia Porchelvi2 and T. Arjun3 Abstract A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. In this paper for a graph G(V;E) each of the vertices v 2 V(G) are assigned by pronic numbers. In this paper, we investigate the pronic heron mean labeling on special cases of generalized Petersen graph P(n;k). Also we described an algorithm to label the vertices for the pronic heron mean labeling for certain disconnected graphs. Keywords Pronic Heron Mean labeling, Generalized Peterson Graph P(n;k), Disconnected Graphs. AMS Subject Classification 05C10. 1,2Department of Mathematics, A.D.M College For Women, (Affiliated to Bharathidasan University), Nagapattinam, Tamil Nadu, India. 3Ideas2IT Technologies, Chennai, Tamil Nadu, India. *Corresponding author: [email protected]; [email protected]; [email protected] Article History: Received 08 January 2021; Accepted 09 February 2021 ©2021 MJM. Contents number of the form n(n + 1). Let G(p;q) be a graph with p ≥ 2.A pronic heron mean labeling[4] of a graph G is a bijec- 1 Introduction.......................................262 tion f : V(G) ! f0;2;6;12;:::; p(p + 1)g such thatp the result- f (u)+ f (v)+ f (u) f (v) 2 Main Theorems....................................263 ing edge labels obtained by f ∗(uv) = d e p 3 2.1 Special cases of P(n;k) ................ 263 ∗ f (u)+ f (v)+ f (u) f (v) or f (uv) = b 3 c for every uv 2 E(G) are 2.2 PHML on Disconnected Graphs.......... 264 all distinct. The present work is aimed to provide pronic 2.3 Algorithm for union of Path graphs Pm [ Pn ... 264 heron mean labeling of some special cases of Generalized 2.4 Algorithm for union of Path Graph and Cycle Graph Peterson graphs namely Durer Graph P(6;2), Mobius Kan- Cm [ Pn ............................ 265 tor GraphP(8;3), Dodecahedren Graph P(10;2), Desargues Graph P(10;3) and Nauru Graph P(12;5). Also we investi- 2.5 Algorithm for union of mK3, m ≥ 2 ......... 265 gate the existence of the same labeling on certain disconnected 2.6 Algorithm for union of C , n ≥ 5 and mK , m ≥ 2 265 n 3 graphs. 2.7 Algorithm for corona product of Comb Graph and Path Graph of (Pn K1) [ Pm, m;n ≥ 3 ....... 266 Definition 1.1. Generalized Peterson Graph For natural numbers n and k, where n > 2k, a Gen- 3 Conclusion........................................266 eralized Peterson graph P(n;k) is the graph whose vertex set References........................................266 is fu1;u2;:::;ung [ f v1;v2;:::;vng and its edge set is fuiui+1;uivi;vivi+k;1 ≤ i ≤ ng, where the subscript arithmetic 1. Introduction is done modulo n using the residues 0;1;2;:::;n − 1. All graphs we consider here are finite, undirected, Note: simple and connected. For standard notations and terminology ) For n = 6 and k = 2, the graph P(6;2) is said to be Durer in graph theory, we follow [1]. We refer [2] for a comprehen- Graph. sive survey on graph labeling and [3] for heron mean labeling. ) For n = 8 and k = 3, the graph P(8;3) is said to be Mobius A pronic number is a number which is the product of two Kantor Graph. Pronic Heron Mean labeling on special cases of generalized Peterson graph P(n;k) and disconnected graphs — 263/266 8 ) For n = 10 and k = 2, the graph P(10;2) is said to be (n + 5)2 + 2 i = 0 <> ∗ 2 Dodecaheran or Dodecahedral Graph. f (vivi+5) = (n + 6) + 2 i = 1: ) n = k = P( ; ) For 10 and 3, the graph 10 3 is said to be :>(n + 3)2 i = 2 Desargues Graph. ) For n = 12 and k = 5, the graph P(12;5) is said to be In the view of the above defined labeling, the Mobius Kantor Nauru Graph. graph admits pronic heron mean labeling. Theorem 2.3. Dodecahedral Graph P(10;2) is a pronic heron 2. Main Theorems mean graph. 2.1 Special cases of P(n;k) Proof. Let fv0;v1;v2;:::;v9g be the inner vertices and Theorem 2.1. Durer Graph P(6;2) admits pronic heron mean fu0;u1;u2;:::;u9g be the outer vertices of P(10;2). labeling. Define a bijection f : V(G) ! fp1; p2;:::; p2ng by 8 Proof. Let fv0;v1;v2;v3;v4;v5g be the inner vertices and p i = 2;3;:::9 ( <> i+1 fu0;u1;u2;u3;u4;u5g be the outer vertices of P(6;2). pi+10 i = 1;2:::9: f (ui) = pi+2 i = 0; ; f (vi) = Define a bijection f : V(G) ! fp1; p2;:::; p2ng by > p20 i = 0 :pi i = 1 f (u ) = p ;i = 0;1;:::;5; f (v ) = p ;i = 0;1;:::;5: i i+1 i 7+i For the above vertex labeling, the edge labeling f ∗ : E(G) ! For the above vertex labeling, the edge labeling f ∗ : E(G) ! f4;5;6;:::; p2ng is defined by f4;5;6;:::; p g is defined by ( 2n (i + 2)2 i = 0;2;3;:::;8; f ∗(u v ) = ∗ 2 i i+1 f (uivi+1) = (i + 2) ;i = 0;1;2;:::;4; (i + 1)(i + 2) i = 1 2 p 2 ∗ n + n + 2 + 2(n) + 2n 2 p 2 f (u0un−1) = d e; ∗ n + n + 6 + 6(n) + 6n 3 f (u0un−1) = d e; ∗ 3 f (vivi+2) = p8+i;i = 0;1;2;3; ( f ∗(v v ) = + i;i = ; ∗ p11+i;i = 1;2;:::;7; i i+4 91 20 0 1; f (vivi+1) = ∗ 2 p16 + 5;i = 0 f (uivi) = (i + 5) − (i + 2);i = 0;1;2;:::5: ∗ f (vivi+8) = 380 − 135i;i = 0;1; In the view of the above defined labeling, the Durer graph 8 2 admits pronic heron mean labeling. >(n + i − 4) + 6;i = 2;3;:::9 ∗ < f (uivi) = 5(i + 9);i = 1: Theorem 2.2. Mobius Kantor Graph P(8;3) admits pronic :>15n + 9;i = 0 heron mean labeling. In the view of the above defined labeling, the Dodecahedral Proof. Let fv0;v1;v2;:::;v7g be the inner vertices and graph admits pronic heron mean labeling. fu ;u ;u ;:::;u g be the outer vertices of P(8;3). 0 1 2 7 Desargues Graph P( ; ) admits pronic heron Define a bijection f : V(G) ! fp ; p ;:::; p g by Theorem 2.4. 10 3 1 2 2n mean labeling. ( pi+10 i = 0;1;2;:::;6: Proof. Let fv0;v1;v2;:::;v9g be the inner vertices and f (ui) = pi+1;i = 0;1;2;:::;7; f (vi) = pi+2 i = 7 fu0;u1;u2;:::;u9g be the outer vertices of P(10;3). Define a bijection f : V(G) ! fp0; p1; p2;:::; p2ng by ∗ For the above vertex labeling, the edge labeling f : E(G) ! 8 p i = 2;3;:::9 f4;5;6;:::; p2ng is defined by <> i+1 f (ui) = pi+2 i = 0; ∗ 2 f (uiui+ ) = (i + 2) ;i = 0;1;2:::;6; > 1 :pi i = 1 2 p 2 ( ∗ n + n + 2 + 2(n) + 2n p i = ; ::: : f (u0un−1) = d e: i+10 1 2 9 3 f (vi) = p20 i = 0 ( 2 ∗ (n + i − 2) + 6 i = 0;1;2:::;6; For the above vertex labeling, the edge labeling f ∗ : E(G) ! f (uivi) = 2 (i + 2) i = 7 f4;5;6;:::; p2ng is defined by ( 2 ( 2 ∗ (n + i + 4) i = 0;1;2;3; ∗ (i + 2) i = 0;2;3;:::;8; f (vivi+3) = f (uivi+1) = (i + 8)2 + 2 i = 4 (i + 1)(i + 2) i = 1 263 Pronic Heron Mean labeling on special cases of generalized Peterson graph P(n;k) and disconnected graphs — 264/266 2 p 2 ∗ n + n + 6 + 6(n) + 6n 2.3 Algorithm for union of Path graphs Pm [ Pn f (u0un−1) = d e; 3 The union graph Pm [ Pn where m;n ≥ 2has the ( vertex set V(Pm [ Pn)= V(Pm) [ V(Pn) with the cardinality (n + + i)2 i = ;:::; ∗ 2 1 6; m + n and edge set E(Pm [ Pn)= E(Pm) [ E(Pn) with cardinal- f (vivi+3) = (n + 7)2 + 4 i = 0 ity q = m + n − 2. Let the path Pm is demonstrated by listing the vertices and the ( 2 ∗ (n + 9) i = 0 edges in order f (vivi+7) = (n + 4 + i)2 + 4 i = 1;2; u1;e1;u2;e2;:::um−1;em−1;um. We name the vertex u1, the active vertex and the vertex um is the end vertex of the edge 8 (n + i − 4)2 + 6;i = 2;3;:::9 em−1: Now the path Pn is demonstrated by listing the ver- <> 0 0 0 ∗ tices and the edges in order v1;e ;v2;e ;:::vm−1;e ;vm. We f (uivi) = 5(i + 9);i = 1: 1 2 m−1 name the vertex v1, the first vertex and the vertex vm is the :>15n + 9;i = 0 0 end vertex of the edge em−1: In the view of the above defined labeling, the Desargues graph Label the vertices in cloclwise direction C. The algorithm admits pronic heron mean labeling. has single pass: it labels the vertices of both the paths Pm and Pn.