Pronic Heron Mean Labeling on Special Cases of Generalized Peterson Graph P(N,K) and Disconnected Graphs

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 ∈ 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) → {0,2,6,12,..., p(p + 1)} such that√ the result- f (u)+ f (v)+ f (u) f (v) 2 Main Theorems...... 263 ing edge labels obtained by f ∗(uv) = d e √ 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 ∈ 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 {u1,u2,...,un} ∪ { v1,v2,...,vn} and its edge set is {uiui+1,uivi,vivi+k,1 ≤ i ≤ n}, 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

 ⇒ 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 {v0,v1,v2,...,v9} be the inner vertices and Theorem 2.1. Durer Graph P(6,2) admits pronic heron mean {u0,u1,u2,...,u9} be the outer vertices of P(10,2). labeling. Define a bijection f : V(G) → {p1, p2,..., p2n} by  Proof. Let {v0,v1,v2,v3,v4,v5} be the inner vertices and p i = 2,3,...9 (  i+1 {u0,u1,u2,u3,u4,u5} 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) → {p1, p2,..., p2n} 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) → {4,5,6,..., p2n} is defined by {4,5,6,..., p } 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  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 {v0,v1,v2,...,v7} be the inner vertices and graph admits pronic heron mean labeling. {u ,u ,u ,...,u } be the outer vertices of P(8,3). 0 1 2 7 Desargues Graph P( , ) admits pronic heron Define a bijection f : V(G) → {p , p ,..., p } by Theorem 2.4. 10 3 1 2 2n mean labeling. ( pi+10 i = 0,1,2,...,6. Proof. Let {v0,v1,v2,...,v9} be the inner vertices and f (ui) = pi+1,i = 0,1,2,...,7; f (vi) = pi+2 i = 7 {u0,u1,u2,...,u9} be the outer vertices of P(10,3). Define a bijection f : V(G) → {p0, p1, p2,..., p2n} by ∗ For the above vertex labeling, the edge labeling f : E(G) →  p i = 2,3,...9 {4,5,6,..., p2n} 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 {4,5,6,..., p2n} 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  (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. At the end of the algorithm, we compute the labels of the Theorem 2.5. Nauru Graph P(12,5) admits pronic heron edges which results the graph, a pronic heron mean graph. mean labeling. Algorithm

Proof. Let {v0,v1,v2,...,v11} be the inner vertices and The parameters of the algorithm are described as follows: {u0,u1,u2,...,u11} be the outer vertices of P(12,5). • i, the index of the vertices ranging from 1 to max{m,n}. Deﬁne a bijection f : V(G) → {p0, p1, p2,..., p2n} by •{u1,u2...,um}, the vertices of Pm, m ≥ 2. •{v1,v2...,vn}, the vertices of Pn, n ≥ 2. f (ui) = pi+1,i = 0,1,2...11; • f (ui), the ”i”th value of the vertex ui. ( • f (vi), the ”i”th value of the vertex vi. pi+12 i = 1,2...11. f (vi) = • pi, the ”i”th pronic number: pi = i(i + 1) for i = p2n i = 0 0,1,2...m + n.

∗ For the above vertex labeling, the edge labeling f : E(G) → i f (m < 2 || n < 2) {4,5,6,..., p2n} is deﬁned by { return ∗ 2 ; f (uiui+1) = (i + 2) ,i = 0,1,2...,10; } n2 + n + 2 + p2(n)2 + 2n let pointersize; f ∗(u u ) = d e; 0 n−1 3 i f (m is greater than n) { pointersize = m; ( 2 ∗ (i + 7) + 10 i = 1,2...,11; } f (uivi) = (n + 3)2 − 13 i = 0 elsei f (n is greater than m) { pointersize = n; ( 2 } ∗ (15 + i) + 2 i = 1,2,...6; f (vivi+5) = 2 else (2n − 3) + 4 i = 0 { pointersize = m; } ( 2 ∗ (16 + i) + 4 i = 1,2,3,4. f or(i=1; i ≤ pointersize;i + +) f (vivi+7) = (2n − 2)2 + 2 i = 0 { if(i ≤ m) In the view of the above deﬁned labeling, the Nauru graph { admits pronic heron mean labeling. f(ui) = pi } 2.2 PHML on Disconnected Graphs if(i ≤ n) Note 2.6. In this section the assignment of the labels are { in both anticlockwise and clockwise directions. Denote the f(vi) = pm+i anticlockwise direction by AC and the clockwise direction by } C. }

264 Pronic Heron Mean labeling on special cases of generalized Peterson graph P(n,k) and disconnected graphs — 265/266

2.4 Algorithm for union of Path Graph and Cycle { Graph Cm ∪ Pn f(ui) = pi+1 The union graph Cm ∪Pn where m,n ≥ 2has the ver- } tex set V(Cm ∪Pn)= V(Cm)∪V(Pn) with the cardinality m+n elseif(i=4) and edge set E(Cm ∪Pn)= E(Cm)∪E(Pn) with cardinality q = { m + n − 1. Let the cycle Cm is demonstrated by listing the ver- f(ui) = pi−1 tices and the edges in order u1,e1,u2,e2,...um−1,em−1,um,em,u1. } We name the vertex u1, the active vertex and the vertex um is } the end vertex of the edge em−1. Now the path Pn is demon- else strated by listing the vertices and the edges in order { 0 0 0 v1,e1,v2,e2,...vm−1,em−1,vm. We name the vertex v1, the ﬁrst f(ui) = pi 0 } vertex and the vertex vm is the end vertex of the edge em−1. Label the vertices in cloclwise direction C. The algorithm has f(vi) = pm+i single pass: it labels the vertices of both the paths Cm and } Pn. At the end of the algorithm, we compute the labels of the } edges which results the graph, a pronic heron mean graph. 2.5 Algorithm for union of mK3, m ≥ 2 Algorithm Given a graph mK3 where m denotes the number of copies of K3. Let the vertex set of mK3 be V = {V1 ∪V2 ∪ The parameters of the algorithm are described as follows: i i i ... ∪ Vm} where Vi = {v1,v2,v3} where i = 1,2,3...,m and • i, the index of the vertices ranging from 1 to i i i the edge set of mK3 be E = {(v1v2) ,(v2v3) ,(v3v1) } where max{m,n}. i = 1,2,3...,m. •{u1,u2,u3,...,um} be the vertices of Cm, m ≥ 3. Label the vertices in cloclwise direction C. The algorithm has •{v ,v ...,v } be the pendant edges attached to the 1 2 n single pass: it labels the vertices of all the mK3 . At the end corresponding vertices of Pn of the algorithm, we compute the labels of the edges which • f (ui), the ”i”th value of the vertex ui. results the graph, a pronic heron mean graph. • f (vi), the ”i”th value of the vertex vi. • pi, the ”i”th pronic number: pi = i(i + 1) for i = 1,2...m + n. Algorithm

i f (m < 3 || n < 2) The parameters of the algorithm are described as follows: { • i, the index of the vertices ranging from 1 to m. i i i return; •{v1,v2,v3}, the vertices of mK3, m ≥ 2. 0 0 } • f (vi), the ”i”th value of the vertex vi. let pointersize; • pi, the ”i”th pronic number: pi = i(i + 1) for i f (m is greater than n) i = 1,2...3m. { pointersize = m; f or(i = 1;i ≤ m;i + +) } { i elsei f (n is greater than m) f (v1) = p3i−2 i { f (v2) = p3i−1 i pointersize = n; f (v3) = p3i } } else { pointersize = m; 2.6 Algorithm for union of Cn, n ≥ 5 and mK3, m ≥ 2 } Given a graph Cn ∪ mK3 where m denotes the num- f or(i=1; i ≤ pointersize;i + +) ber of copies of K3 and n denote the number of vertices of Cn. { Let the cycle Cm is demonstrated by listing the vertices and the if(m==4) edges in order u1,e1,u2,e2,...um−1,en−1,un,en,u1. We name { the vertex u1, the active vertex and the vertex un is the end ver- if(i=1 k i = 2) tex of the edge en−1. Let the vertex set of mK3 be V = {V1 ∪ j j j { V2 ∪ ... ∪Vm} where Vj = {v1,v2,v3} where j = 1,2,3...,m j i j f(ui) = pi and the edge set of mK3 be E = {(v1v2) ,(v2v j) ,(v3v1) } } where j = 1,2,3...,m. elseif(i=3) Label the vertices in clockwise direction C. The algorithm

265 Pronic Heron Mean labeling on special cases of generalized Peterson graph P(n,k) and disconnected graphs — 266/266

has single pass: it labels the vertices of all the Cn and the it } labels all the evrtices of mK3 . At the end of the algorithm, let pointersize; we compute the labels of the edges which results the graph, a i f (m is greater than n) pronic heron mean graph. { pointersize = m; } Algorithm elsei f (n is greater than m) The parameters of the algorithm are described as follows: { • i, the index of the vertices ranging from 1 to n. pointersize = n; • j, the index of the vertices ranging from 1 to m. } •{u1,u2,u3,...,un} be the vertices of Cn, n ≥ 3. else j j j { •{v1,v2,v3}, the vertices of mK3, m ≥ 2. • pi, the ”i”th pronic number: pi = i(i + 1) for pointersize = m; i = 1,2...n + m. } for(i=1; i ≤ pointersize;i + +) i f (m < 2 || n < 5) { { f (vi) = p2i−1 0 return; f (vi) = p2i } f (ui) = p2n+i for(i=1; i ≤ n;i + +) } { f (ui) = pi } 3. Conclusion for(i=1; j ≤ m; j + +) { In this paper, the results for few special graphs are j proved that they admit pronic heron mean labeling. It is f (v1) = pn+3 j−2 j possible to investigate similar results for other families of f (v2) = pn+3 j−1 graphs. The authors are highly thankful to the anonymous j f (v3) = pn+3 j referees for their valuable suggestions. } References 2.7 Algorithm for corona product of Comb Graph [1] Harary. F, Graph Theory, Narosa Publishing House, New and Path Graph of (Pn K1) ∪ Pm, m,n ≥ 3 Delhi, 1988. The comb graph, denoted by Pn K1 is deﬁned as [2] Joseph A. Gallion, A Dynamic Survey of Graph Labeling, the corona product of the path Pn and K1 . It has 2n vertices The Electronic Journal of Combinatorics, Vol. no. 19, and 2n − 1 edges. Let {u0,u1,...,un−1} be the path Pn with n 2012. 0 0 0 [3] vertices and{v1,v2,...,vn} be the pendant edges attached to Sambath Kumar. R, Narasimhan. G, Nagaraja. K. M, the corresponding vertices of Pn. Let {u1,u2,u3,...,um} be Heron Mean Labeling of Graphs, International Journal the vertices of Pm, m ≥ 3. Algorithm for Comb Graph: of Recent Scientiﬁc Research, 8(2017), 19808-19811. The parameters of the algorithm are described as follows: [4] Sophia Porchelvi. R, Akila Devi. S, Pronic Heron Mean • i, the index of the vertices ranging from 1 to 1 to Graphs, The International Journal of Analytical and max{m,n}. Experimental Modal Analysis, Volume XII, Issue III, •{v1,v2...,vn}, the vertices of Pn, n ≥ 3. March/2020, 2207-2212. 0 0 0 •{v1,v2,...,vn}, the vertices attached to the corresponding vertices of Pn. ????????? •{u1,u2,u3,...,um}, the vertices of Pm, m ≥ 3. ISSN(P):2319 − 3786 • f (vi), the ”i”th value of the vertex vi. 0 0 • f (v ), the ”i”th value of the vertex v . Malaya Journal of Matematik i i ISSN(O):2321 − 5666 • f (ui), the ”i”th value of the vertex ui. ????????? • pi, the ”i”th pronic number: pi = i(i + 1) where i = 1,2...2n + m.

i f (m < 3 || n < 2) { return;

266