# On Square Sum Labeling of Two Families of Petersen Graphs

Authors: Zhiqiang Zhang; Muhammad Naeem; Abeera Tariq; Weidong Zhao
Journal: Journal of Mathematics
 (2022)

---

## Abstract

A labeling on a graphG with n vertices and m edges is called square sum if there exists a bijection f:VG⟶0,1,2,3,…,n−1 such that the function f∗:EG⟶N defined by f∗st=fs2+ft2, for all st∈EG, is injective. A graph G having a square sum labeling is called square sum. In this study, we have investigated the square sum labeling of generalized Petersen graph and double generalized Petersen graph.


---

## Body

## 1. Introduction

In [1], Germina et al. derived square sum labeling for basic graphs such as trees and cycles. Among various labeling methods that have evolved since 1960, one of the significant labeling methods is square sum. The vast history of the sum of squares of the integers motivated them to study and define this labeling in the particular graphs, and they named it square sum graphs. Many problems in real life and abstract thoughts, which are based on several conditions and with practical implications, can be studied by understanding modeling those problems by graphs, and this problem is one of the uses of graphs in handling and attempting to solve a problem of number theory. A graph which admits square sum labeling is called a square sum graph. In [1], Germina et al. proved that the complete graph Kn is square sum iff n≤7 and the other graphs which they proved to be square sum are trees, unicyclic graphs, mCn, and cycles’ chord; the graphs obtained by connected two copies of cycle Cn passes through the path Pk, a path union of k copies of Cn, and the path is P2.LetG be a graph and p and q be its vertices and edges, respectively. A labeling on the vertices G can also be called a function with domain VG. An injective function with domain VG and range 0,1,2,…,p−1 is called square sum labeling if the induced function of f, namely, the function f∗ with domain EG and defined as f∗st=fs2+ft2, for all st∈EG, is also injective. The square sum graphs are the graphs having a square sum labeling. We would also like to mention that not all graphs are square sum graphs. For instance, the complete graphs Kn, for n≥6, are not square sum graphs.In number theory form, the statement that “a numbern can be written as a sum of two squares integers” is equivalent to “every prime of the form 4k+3 occurs an even number of times in the prime factorization of n.” The edge labels in square sum labeling is of the form n=a2+b2. So, a number n cannot be an edge label of a square sum graph if in its prime factorization a prime of the form 4k+3 (if it exists) occurs an odd number of times. Seoud and Al-Harere [2] proved many necessary conditions of square sum graphs, and they showed that 2Cn, P2n, and C2n are square sum graphs. In [3], Huilgol and Sriram prove that if G1 and G2 are square sum, then G1∪G2∪G3 is also square sum, where G3 is a set of isolated vertices, for more literature review (see [2–5]).


## 2. Main Results

In this section, we will present our results of this article.Theorem 1.
The generalized Petersen graphPn,k is a square sum graph, for all n≥5.Proof.
SupposeG≅Pn,k be the generalized Petersen graph, where 2≤k≤⌊n/2⌋. We shall show that G is square sum graph by defining a square sum labeling on G. For this purpose, we shall consider two cases one is when n is even and other when n is odd.
So, supposef:VG⟶0,1,…,VG−1 be a function defined as follows.Case 1.
Whenn is even.
The labeling ofyi for 2≤i≤n/2−1 is(1)fyi=4i−2.
The labeling offy0 and fy1 is fixed:(2)fy0=1,fy1=2.
The labeling ofyi for n/2+1≤i≤n−1 is(3)fyi=2n−3−4i−n2−1,fyi=2n−1,i=n2.
The labeling offx0 and fx1 is fixed:(4)fx0=0,fx1=3.
The labeling of verticesxi for 2≤i≤n/2−1 is(5)fxi=4i−1,fxi=2n−2,i=n2.
The labeling ofxi for n/2+1≤i≤n−1 is(6)fxi=2n−4−4i−n2−1.
The labeling ofxi is(7)fxi=2n+4n2−4i=2n+2n−4i=4n−4i.
This labeling is explained in Figure1.
Now, we shall discuss the weights of the edges induced by the above labeling of vertices ofG. The weight of the edge uv∈EG will be denoted as f∗uv=fu2+fv2. So, the induced labeling of the edge xiyi for 0≤i≤n−1 is given as follows.
The labeling ofxiyi, for 2≤i≤n/2−1, is(8)f∗xiyi=fxi2+fyi2;2≤i≤n2−1,=4i−12+4i−22=32i2−24i+5.
The labeling ofx0y1 and x1y1 is fixed:(9)fx0y1=1,fx1y1=13.
The square sum labeling of edgexiyi, for i=n/2, is(10)f∗xiyi=fxi2+fyi2=2n−12+2n−22=4n2−4n+1+4n2−8n+4=8n2−12n+5.
The square sum labeling ofxiyi, for n/2+1≤i≤n−1, is(11)f∗xiyi=fxi2+fyi2=2n−4−4i−n2−12+2n−3−4i−n2−12=32n2−64ni+32i2+8n−8i+1.
The square sum labeling ofyiyi+1, for 2≤i≤n/2−2, is(12)f∗yiyi+1=fyi2+fyi+12=4i−22+4i+1−22=16i2−16i+4+16i2+16i+4=32i2+8.
The following labels are fixed:(13)f∗y0y1=5,f∗y1y2=40,f∗yn−1y0=26.
The square sum labeling of the edgesyiyi+1, for i=n/2−1, is(14)f∗yiyi+1=fyi2+fyi+12=4i−22+2n−12=16i2−16i+4+4n2−4n+1.
The labeling of the edgesyiyi+1, for i=n/2, is(15)f∗yiyi+1=fyi2+fyi+12=2n−12+2n−32=8n2−16n+10.
The square sum labeling of the edgesyiyi+1, for n/2+1≤i≤n−2, is(16)f∗yiyi+1=fyi2+fyi+12=2n−3−4i−n2−12+2n−3−4i−n2−12=32n2−64ni−16n+32i2+16i+10.
The labeling of edges ofxixi+1, for 1≤i≤n/2−1, is(17)f∗xixi+k=fxi2+fxi+k2=4i−12+4i+k−12=32i2−16i2+32ik+16k2−8k+2.
The labeling of the edgesxixi+1, for 1≤i≤n/2−1andi+k=n/2, is(18)f∗xixi+k=fxi2+fxi+k2=4i−12+2n−12=16i2−8i+4n2−4n+2.
The labeling ofxixi+1, for if1≤i≤n/2−1andi+k>n/2, is(19)f∗xixi+k=fxi2+fxi+k2=4i−12+fxi+k2=4i−12+2n−4−4i+k−n2−12=16i2−8i+1+4n−4k−4i=16i2−12i+4n−4k+1.
The labeling ofxixi+k, for i=n/2, is(20)f∗xixi+k=fxi2+fxi+k2=2n−22+2n−4−4i−n2−12=20n2+16i2+16n2−32ni+8n−4.
The labeling ofxixi+k, for n/2+1≤i≤n−1andi+k≤n−1, is(21)f∗xixi+k=fxi2+fxi+k2=2n−4−4i−n2−12+2n−4−4i+k−n2−12=16n2−32ni+16i2+16i2+16n2−32ni−32nk+32ik+16i2+16k2=32n2−64ni+32i2+32nk+16i2+16k2.
The labeling ofxixi+k, for n/2+1≤i≤n−1andi+k=n=0modn, is(22)f∗xixi+k=fxi2+fxi+k2=2n−4−4i−n2−12+0=16n2+16i2−32ni−32nk+32ik+16k2.
The labeling ofxixi+k, for n/2+1≤i≤n−1andi+k>nandsayi+k=jandmodn,i≤j≤n/2−2, is(23)f∗xixi+k=fxi2+fxi+k2=2n−4−4i−n2−12+4j−12=16n2+16i2−32ni−32nk+32ik+16k2+16j2−8j+1.
From the computations of all the induced weights of the edges, we can easily see that they are distinct, so the graph is square sum forn even.Figure 1
Square sum labeling of Petersen graphP12,2.Case 2.
Whenn is odd.
SupposeG≅Pn,k be the generalized Petersen graph, where 2≤k≤⌊n/2⌋ and n is odd. We shall show that G is square sum graph by defining a square sum labeling on G. So, let ϕ:VG⟶0,1,…,VG−1 be a function defined as follows.
The labeling ofyn−1,yn−2…yn−6 is given below:(24)fyn−1=2,fyn−2=3,fyn−3=5,fyn−4=8,fyn−5=10,fyn−6=12.
These are the fixed numbers of the above labeling.
The labeling of vertices ofy0 is(25)fy0=2n−1.
The labeling of verticesyi for 1≤i≤n/2 is(26)fyi=2n−1−2i.
The labeling of verticesxn−1,xn−2…,xn−6 is given below:(27)fxn−1=1,fxn−2=4,fxn−3=6,fxn−4=7,fxn−5=9,fxn−6=11.
These are the fixed numbers of the above labeling:(28)fx0=0.
The labelingxi for 1≤i≤n−7 is(29)fxi=2n−2i.
This labeling is explained in Figure2.
From all these computations of the weights of the edges, we can easily see that they are all distinct, that is, the induced labeling functionf∗ is injective just the same way as in Case 1. This shows that f is a square sum labeling, and therefore, Pn,k is a square sum graph.Figure 2
Square sum labeling of Petersen graph P [9,3].


### 2.1. Square Sum Labeling of Double Generalized Petersen Graph

The concept of double generalized Petersen graph was introduced by Zhou and Feng in 2012 (see [6]), where the automorphism group of these graph was characterized. Double generalized Petersen graph DPn,k is defined as the graph with vertex and edge set, as in Figure 3, particularly, for k=1.Figure 3
Generalized double Petersen graphDP10,1.In the next theorem, we have proved that the double generalized Petersen graph is a square sum graph for a particular case whenk=1.Theorem 2.
SupposeDPn,1 be the double generalized Petersen graph; then, there exists square sum labeling of DPn,1, for all n≥5.Proof.
SupposeG≅DPn,1 be the double generalized Petersen graph, where 0≤i≤n−1. We want to show that G is a square sum graph by defining a square sum labeling on G. So, suppose f:VG⟶0,1,…,VG−1 be a function defined as follows.
The labeling ofai vertices, for 0≤i≤n−1, is(30)fai=4i.
The labeling ofbi, for 0≤i≤n−1, is(31)fbi=4i+1.
The labeling ofci, for 0≤i≤n−1, is(32)fci=4i+2.
The labeling ofdi, for 0≤i≤n−1, is(33)fdi=4i=3.
This above labeling is explained in Figure4.
Now, we shall discuss the weights of the edges induced by the above labeling of vertices ofG. The weight of the edge uv∈EG will be denoted as f∗uv=fu2+fv2. So, the induced labeling of the edge aibi, for 0≤i≤n−1, is given as(34)f∗aibi=fai2+fbi2=4i2+4i+12=32i2+8i+1.
The labeling ofcidi, for 0≤i≤n−1, is(35)f∗cidi=fci2+fdi2=4i+22+4i+32=32i2+40i+13.
The labeling ofbici+1, for 0≤i≤n−2, is(36)f∗bici+1=fbi2+fci+12=4i+12+4i+1+22=16i2+8i+16i+12+16i+1+5.
The labeling ofaiai+1, for 0≤i≤n−2, is(37)f∗aiai+1=fai2+fai+12=4i2+4i+12=16i2+4i+42=16i2+16i2+32i+16=32i2+32i+16.
The labeling ofdidi+1, for 0≤i≤n−2, is(38)f∗didi+1=fdi2+fdi+12=4i+32+4i+1+32=16i2+24i+9+16i2+56i+49=32i2+80i+58.
The labeling of the edgesbici, for 0≤i≤n−2, is(39)f∗bici=fbi2+fci2=4i+12+4i+22=i=n−1,i=0=4n−1+1+40+22=4n−4+12+40+22=16n2−24n+13.
The labeling of the edgesbi+1ci, for 0≤i≤n−2, is(40)f∗bi+1ci=fbi+12+fci2=4i+12+4i+22=4i+1+12+4i+22=4i+4+12+4i+22=4i+4+12+4i+22=4i+52+4i+22=16i2+40i+25+16i2+16i+4=32i2+56i+29.
The labeling of the edgesan−1a0, for 0≤i≤n−2, is(41)f∗an−1a0=fan−12+fa02=4i2+4i2=4n−12+402=4n−42=16n2+32n+16.
The labeling of the edgesdn−1d0, for 0≤i≤n−2, is(42)f∗dn−1d0=fdn−12+fd02=4i+32+4i+32=4n−1+32+40+32=4n−4+32+40+32=4n−12+32=16n2−8n+10.
From all these computations of the weights of the edges, we can easily see that they are all distinct, that is, the induced labeling functionf∗ is injective. This shows that f is a square sum labeling, and therefore, DPn,1 is a square sum graph.
In the next theorem, we prove that “the double generalized Petersen graphDPn,2 is a square sum graph for a particular case when k=2.”Figure 4
Square sum labeling of double generalized Petersen graphDP10,1.Theorem 3.
Double generalized Petersen graphDPn,2 is square sum for n≥5.Proof.
LetG≅DPn,2 be the double generalized Petersen graph. For this purpose, we shall consider two cases one is when n is even and other n is odd.Case 3.
Whenn is even.
In this case, we define a labelingf:VG⟶0,1,2,3…VG−1, which will show that this labeling is a square sum labeling. Then, we discuss about the weights of the edges induced by the following labeling.
The labeling of the outer edges:(43)fa0=0,fan/2=1,fan−1=2,fan/2+1=2.
The labeling of the verticesai, for 1≤i≤n/2, is(44)fai=2i−1+3,fan/2−1=n−1.
The labeling of the verticesai, for n/2+2≤i≤n−1, is(45)fai=n−2−2i−n2−1.
The labeling of inner vertices ofbi′s is(46)fb0=n,fbn/2=n+1.
The labeling of the verticesbi, for 1≤i≤n/2, is(47)fbi=n+2i−1+3,fbn/2+1=2n−2,fbn/2−1=2n−2,fbn/2+2=2n−4.
The labeling of the verticesbi, for n/2+2≤i≤n−1, is(48)fbi=2n−2−2i−n2−1,fbn/2−2=2n−3.
The labeling of inner vertices ofci′s is(49)fc0=2n,fcn/2=2n+1.
The labeling ofci, for 1≤i≤n/2, is(50)fci=2n+2i−1+3,fcn−1=2n+2,fcn−2=2n+4,fcn/2−2=3n−3,fcn/2−1=3n−1,fcn/2+1=3n−2,fcn/2+2=3n−4.
The labeling ofci, for n/2+2≤i≤n−1, is(51)fci=3n−2−2i−n2−1.
The labeling of the inner vertices ofdi′s is(52)fd0=3n,fdn/2−1=4n−1,fdn−1=3n+2,fd1=3n+3.
The labeling of the verticesdi, for 1≤i≤n/2, is(53)fdi=3n+2i−1+3,fdn/2+1=4n−2,fdn/2=3n+1.
The labeling ofdi, for n/2+2≤i≤n−1, is(54)fdi=4n−2−2i−n2−1.
This labeling is explained in Figure5.
The labeling of outer edges ofai and inner vertices of bi, for 1≤i≤n−1,i≠0,i≠n/2, is(55)f∗aibi=fai2+fbi2=2i−1+32+n+2i−1+32=8i2+8i+n2+4in+2n+2.
The labeling of inner vertices ofci and inner edges of bi, for 1≤i≤n−1,i≠0,i=n/2, is(56)f∗cidi=fci2+fdi2=2n+2i−1+32+3n+2i−1+32=13n2+20in+10n+8i2+8i+2,f∗cn/2+1dn/2+1=fcn/2+12+fdn/2+12=3n−22+4n−22=25n2−28n+8.
The labeling of the edgesaibi, for n/2+2≤i≤n−1, is(57)f∗aibi=fai2+fbi2=n−2−2i−n2−12+2n−2−2i−n2−12=13n2−20in+8i2.
The labeling of the edgescidi, for n/2+2≤i≤n−1, is(58)f∗cidi=fci2+fdi2=3n−2−2i−n2−12+4n−2−2i−n2−12=41n2−36in+8i2.
The labeling of the edgesbici+2, for 1≤i≤n/2, is(59)f∗bici+2=fbi2+fci+22n2+1≤i≤n−3=n+2i−1+32+2n+2i+2−1+32=5n2+8i2+12in+22n+24i+26.
The labeling ofcn/2+1bi+2 is given below:(60)f∗cn/2+1bi+2=fcn/22+fbi+22=3n−22+2n−2−2i−n2−12=3n−22+2n−2−2i+2−n2−12=18n2+4i2−36n+16i−12in+20.
The labeling of the edgesbi+2ci, for n/2+1≤i≤n−3, is(61)f∗bi+2ci=fbi+22+fci2=2n−2−2i−n2−12+3n−2−2i−n2−12=3n−2i+42+4n−2i2=25n2−28in−24n+8i2+16i+16.
The labeling of the edgesa0b0 is given below:(62)f∗a0b0=fa02+fb02=02+n2=n2.
The labeling of the edgesan/2bn/2 is given below:(63)f∗an/2bn/2=fan/22+fn22=12+n+12=n2+2n+2.
The labeling of the edgesc0d0 is given below:(64)f∗c0d0=fc02+fd02=2n2+3n2=4n2+9n2=13n2.
The labeling of edgescn/2dn/2 is given below:(65)f∗cn/2dn/2=fcn/22+fdn/22=2n+12+3n+12=13n2+10n+2.
The labeling of vertices, forbn/2−2cn/2, is given below:(66)f∗bn/2−2cn/2=fbn/2−22+fcn/22=2n−32+2n+12=8n2−8n+10.
The labeling of edgesbn/2−2cn/2 is given below:(67)f∗bn/2−1cn/2+1=fbn/2−12+fcn/2+12=2n−12+3n−22=13n2−16n+5.
The labeling of edges and vertices, forbn/2cn/2+2, is given below:(68)f∗bn/2cn/2+2=fbn/22+fcn/2+22=n+12+3n−42=10n2−22n+17.
The labeling of verticesbici+2, for n/2+1≤i≤n−3, is(69)f∗bici+2=fbi2+fci+22=2n−22+3n−2−2i−n2−12=4n2−8n+4+3n−2−2i+2−n2−12=4n2−8n+4+4n−2i−42=20n2−40n−16in+4i2+16i+20.
The labeling of verticesbn−2c0 is given below:(70)f∗bn−2c0=fbn−22+fco2=n+42+2n2=5n2+8n+16.
The labeling of the edgesbn−1c1 is given below:(71)f∗bn−1c1=fbn−12+fc12=n+22+2n+2i−1+32,i=1=n2+4n+4+2n+32=5n2+16n+13.
The labeling of the edgesbi+2ci, for 1≤i≤n/2−3, is(72)f∗bi+2ci=fbi+22+fci2=n+2i−1+32+2n+2i−1+32=n+2i+52+2n+i+22=5n2+5i2+8in+18n+24i+29.
The labeling of the edgesc0b2 is given below:(73)f∗c0b2=fco2+fb22=2n2+n+2i−1+32=2n2+n+21+32=2n2+n+52=5n2+10n+25.
The labeling of the verticescn−1b1 is given below:(74)f∗cn−1b1=fcn−12+fb12=2n+22+n+2i−1+32=4n2+8n+4+n+32=4n2+8n+4+n2+6n+9=5n2+14n+13.
The labeling of the verticescn−2b0 is given below:(75)f∗cn−2b0=fcn−22+fb02=2n+42+n2=4n2+16n+16+n2=5n2+16n+16.
The labeling of the edgesbn/2cn/2−2 is given below:(76)f∗bn/2cn/2−2=fbn/22+fcn/2−22=n+12+3n−32=n2+2n+1+9n2−18n+9=10n2−16n+10.
The labeling of the edgesbn/2+2cn/2 is given below:(77)f∗bn/2+2cn/2=fbn/2+22+fcn/22=2n−42+2n+12=8n2−12n+17.
The labeling of the edgesbn/2cn/2+2 is given here:(78)f∗bn/2cn/2−2=fbn/22+fcn/2−22=n+12+3n−32=n2+2n+1+9n2−18n+9=10n2−16n+10.
The labeling of the edgesbn/2+1cn/2−1 is given below:(79)f∗bn/2+1cn/2−1=fbn/2+12+fcn/2−12=2n−22+3n−12=4n2−8n+4+9n2−6n+1=13n2−14n+5.
The labeling of the edgesa0a1 is given below:(80)f∗a0a12=fa02+fa12=02+2i−1+32,i=1=02+20+32=02+32=9.
The labeling of the edgesaiai+1, for 1≤i≤n/2−2, is(81)f∗aiai+1=fai2+fai+12=2i−1+32+2i+1−1+32=2i+12+2i+32=8i2+16i+10.
The labeling of the edgesan/2−1an/2 is given below:(82)f∗an/2−1an/22=fan/2−12+fan/22=n−12+12=n2−2n+2.
The labeling of edges with vertices having labelsan/2an/2+1 is(83)f∗an/2an/2+1=fan/22+fan/2+12=12+n−22=n2−4n+5.
The labeling of the edgesan−1a0 is here:(84)f∗an−1a0=fan−12+fa02=22+02=4.
The square sum labeling of the edgesaiai+1, for n/2+2≤i≤n−2, is(85)f∗aiai+1=fai2+fai+12,n2+2≤i≤n−2=n−2−2i−n2−12+n−2−2i+1−n2−12=2n−2i2+2n−2i−22=4n2−8in+4i2+4n2−8in−8n+8i+4i2+4=8n2−16in+8i2−8n+8i+4.
The labeling ofd0d1 is the square sum:(86)f∗d0d1=fd02+fd12=3n2+3n+32=9n2+9n2+18n+9=18n2+18n+9.
The labeling of the edges ofd0d1 is the square sum:(87)f∗didi+1=fdi2+fdi+12=3n+2i−1+32+3n+2i+1−1+32=3n+2i+12+3n+2i+32=9n2+12in+18n+4i2+12i+9+9n2+12in+42+4i+6n+1=18n2+24in+24n+8i2+16i+10.
The labeling of the edgesdn/2−1dn/2 is given below:(88)f∗dn/2−1dn/2=fdn/2−12+fdn/22=4n−12+3n+12=16n2−8n+1+9n2+6n+1=25n2−2n+2.
The labeling of the edgesdn/2dn/2+1 is the square sum:(89)f∗dn/2dn/2+1=fdn/22+fdn/2+12=n+12+4n−22=9n2+6n+1+16n2−16n+5=25n2−10n+5.
The labeling of the edgesdn−1d0 is the square sum:(90)f∗dn−1d0=fdn−12+fd02=3n+22+3n2=9n2+6n+6n+4+9n2=18n2+12n+4.
The labeling of the edgesdidi+1 is the square sum:(91)f∗didi+1=fdi2+fdi+12=4n−2−2i−n2−12+4n−2−2i+1−n2−12=5n−2i2+5n−2i−22=25n2−20in+4i2+25n2−20in−20n+4i2+8i+4=50n2−40in+8i2+8i−20n+4.
From all these computations of the weights of the edges, we can easily see that they are all distinct, that is, the induced labeling functionf∗ is injective. This shows that f is a square sum labeling, and therefore, DPn,2 is a square sum graph.
Here, we prove Case2, when n is odd.
So, this graph shows square sum labeling. LetG≅DPn,2 be the double generalized Petersen graph, where 1≤i≤n−1. We shall show that G is square sum graph by defining a square sum labeling on G.Figure 5
Square sum labeling ofDP12,2.Case 4.
Whenn is odd.
For this case, we define a labelingf:VG⟶0,1,2,3…VG−1, which will show that this labeling is a square sum labeling.
The labeling ofa0 is(92)fa0=1.
The labeling ofb0 is(93)fb0=n+2.
The labeling ofc0 is(94)fc0=n+1.
The labeling ofd0 is(95)fd0=0.
The labeling ofai, for 1≤i≤n−1, is(96)fai=i+1.
The labeling ofbi, for 1≤i≤n−1, is(97)fbi=n+2+i.
The labeling ofci, for 2≤i≤n−1, is(98)fci=3n+i.
The labeling ofdi, for 1≤i≤n−1, is(99)fdi=2n+i+1.
The labeling in this case is explained in Figure6.
The square sum labeling of the edgea0b0 is(100)f∗a0b0=fa02+fb02=12+n+22=n2+4n+5.
The labeling of edges with vertices having labelsaibi, for 1≤i≤n−1, is(101)f∗aibi=fai2+fbi2=i+12+n+i+22=n2+2i2+6i+4n+2in+5.
The square sum labeling of the labelsc0d0 is(102)f∗c0d0=fc02+fd02=n+12+02=n2+2n+1.
The square sum labeling of the labelsc1di is(103)f∗c1di=fc12+fdi2=3n+12+2n+i+12=13n2+10n+i2+4in+2i+2.
The square sum labeling of the labelscidi, for 2≤i≤n−1, is(104)f∗cidi=fci2+fdi2=3n+i2+2n+i+12=13n2+10in+2i2+4n+2i+1.
The square sum labeling ofdid0, for i=n−1, is(105)f∗did0=fdi2+fd02=2n+i+12+02=2n+n−1+12+02=3n2=9.
The square sum labeling ofaia0, for i=n−1, is(106)f∗aia0=fai2+fa02=i+12+12=n−1+12+12=n2+12=n2+1.
The labeling of the edgesaiai+1, for 0≤i≤n−2, is given below:(107)f∗aiai+1=fai2+fai+12=i+12+i+1+12=i+12+i+22=2i2+6i+5.
The labeling of the edgesdidi+1, for 0≤i≤n−2, is given below:(108)f∗didi+1=fdi2+fdi+12=2n+i+12+2n+i+1+12=8n2+8in+12n+2i2+6i+5.
The square sum labeling of the edgesc0bi, for i=n−2, is(109)f∗c0bi=fc02+fbi2=n+12+n+i+22=n+12+n+2+n−22=n+12+n+2+n−22=n+12+2n2=5n2+2n+1.
The square sum labeling of the edgesb0ci, for i=n−2, is(110)f∗b0ci=fb02+fci2=n+22+3n+i2=n+22+3n+n−22=n2+4n+4+16n2+4−16n=17n2−12n+8.
The labeling of the edgesbici is(111)f∗=bici=fbi2+fci2=n+2+i2+3n+i2=n+2+12+3n+i2;bi=n+2+i⇒i=1=n+32+3n+n−12;ci=3n+i⇒i=n−1=n+32+4n−12=17n2−2n+10.
The square sum labeling of the edgesc0bi, for i=2, is(112)f∗c0bi=fc02+fbi2=n+12+n+i+22=2n2+2in+6n+i2+4i+5.
The square sum labeling of the edgescibi+2, for 1≤i≤n−3, is(113)f∗cibi+2=fci2+fbi+22=3n+i2+n+2+i2=10n2+8n+8in+2i2+8i+16.
The square sum labeling of the edgescibi, for 0≤i≤n−3, is(114)f∗cibi=fci2+fbi2=3n+i2+n+2+i2;ci⇒i=i+2=3n+i+22+n+2+i2=10n2+8in+16n+2i2+8i+8.
The square sum labeling of the edgescibi, for 0≤i≤n−3, is(115)f∗cibi=fci2+fbi2=3n+i2+n+2+i2;ci⇒i=i+2=3n+i+22+n+2+i2=10n2+8in+16n+2i2+8i+8.
The labeling of the edgesc1bi is given below:(116)f∗c1bi=fc12+fbi2=3n+12+n+2+i2;ci⇒i=n−1=3n+12+n+2+n−12=3n+12+2n+12=13n2+10n+2.
From all these computations of the weights of the edges, we can easily see that they are all distinct, that is, the induced labeling functionf∗ is injective. This shows that f is a square sum labeling, and therefore, DPn,2 is a square sum graph.Figure 6
Square sum labeling of double generalized Petersen graph [11,2].


## 2.1. Square Sum Labeling of Double Generalized Petersen Graph

The concept of double generalized Petersen graph was introduced by Zhou and Feng in 2012 (see [6]), where the automorphism group of these graph was characterized. Double generalized Petersen graph DPn,k is defined as the graph with vertex and edge set, as in Figure 3, particularly, for k=1.Figure 3
Generalized double Petersen graphDP10,1.In the next theorem, we have proved that the double generalized Petersen graph is a square sum graph for a particular case whenk=1.Theorem 2.
SupposeDPn,1 be the double generalized Petersen graph; then, there exists square sum labeling of DPn,1, for all n≥5.Proof.
SupposeG≅DPn,1 be the double generalized Petersen graph, where 0≤i≤n−1. We want to show that G is a square sum graph by defining a square sum labeling on G. So, suppose f:VG⟶0,1,…,VG−1 be a function defined as follows.
The labeling ofai vertices, for 0≤i≤n−1, is(30)fai=4i.
The labeling ofbi, for 0≤i≤n−1, is(31)fbi=4i+1.
The labeling ofci, for 0≤i≤n−1, is(32)fci=4i+2.
The labeling ofdi, for 0≤i≤n−1, is(33)fdi=4i=3.
This above labeling is explained in Figure4.
Now, we shall discuss the weights of the edges induced by the above labeling of vertices ofG. The weight of the edge uv∈EG will be denoted as f∗uv=fu2+fv2. So, the induced labeling of the edge aibi, for 0≤i≤n−1, is given as(34)f∗aibi=fai2+fbi2=4i2+4i+12=32i2+8i+1.
The labeling ofcidi, for 0≤i≤n−1, is(35)f∗cidi=fci2+fdi2=4i+22+4i+32=32i2+40i+13.
The labeling ofbici+1, for 0≤i≤n−2, is(36)f∗bici+1=fbi2+fci+12=4i+12+4i+1+22=16i2+8i+16i+12+16i+1+5.
The labeling ofaiai+1, for 0≤i≤n−2, is(37)f∗aiai+1=fai2+fai+12=4i2+4i+12=16i2+4i+42=16i2+16i2+32i+16=32i2+32i+16.
The labeling ofdidi+1, for 0≤i≤n−2, is(38)f∗didi+1=fdi2+fdi+12=4i+32+4i+1+32=16i2+24i+9+16i2+56i+49=32i2+80i+58.
The labeling of the edgesbici, for 0≤i≤n−2, is(39)f∗bici=fbi2+fci2=4i+12+4i+22=i=n−1,i=0=4n−1+1+40+22=4n−4+12+40+22=16n2−24n+13.
The labeling of the edgesbi+1ci, for 0≤i≤n−2, is(40)f∗bi+1ci=fbi+12+fci2=4i+12+4i+22=4i+1+12+4i+22=4i+4+12+4i+22=4i+4+12+4i+22=4i+52+4i+22=16i2+40i+25+16i2+16i+4=32i2+56i+29.
The labeling of the edgesan−1a0, for 0≤i≤n−2, is(41)f∗an−1a0=fan−12+fa02=4i2+4i2=4n−12+402=4n−42=16n2+32n+16.
The labeling of the edgesdn−1d0, for 0≤i≤n−2, is(42)f∗dn−1d0=fdn−12+fd02=4i+32+4i+32=4n−1+32+40+32=4n−4+32+40+32=4n−12+32=16n2−8n+10.
From all these computations of the weights of the edges, we can easily see that they are all distinct, that is, the induced labeling functionf∗ is injective. This shows that f is a square sum labeling, and therefore, DPn,1 is a square sum graph.
In the next theorem, we prove that “the double generalized Petersen graphDPn,2 is a square sum graph for a particular case when k=2.”Figure 4
Square sum labeling of double generalized Petersen graphDP10,1.Theorem 3.
Double generalized Petersen graphDPn,2 is square sum for n≥5.Proof.
LetG≅DPn,2 be the double generalized Petersen graph. For this purpose, we shall consider two cases one is when n is even and other n is odd.Case 3.
Whenn is even.
In this case, we define a labelingf:VG⟶0,1,2,3…VG−1, which will show that this labeling is a square sum labeling. Then, we discuss about the weights of the edges induced by the following labeling.
The labeling of the outer edges:(43)fa0=0,fan/2=1,fan−1=2,fan/2+1=2.
The labeling of the verticesai, for 1≤i≤n/2, is(44)fai=2i−1+3,fan/2−1=n−1.
The labeling of the verticesai, for n/2+2≤i≤n−1, is(45)fai=n−2−2i−n2−1.
The labeling of inner vertices ofbi′s is(46)fb0=n,fbn/2=n+1.
The labeling of the verticesbi, for 1≤i≤n/2, is(47)fbi=n+2i−1+3,fbn/2+1=2n−2,fbn/2−1=2n−2,fbn/2+2=2n−4.
The labeling of the verticesbi, for n/2+2≤i≤n−1, is(48)fbi=2n−2−2i−n2−1,fbn/2−2=2n−3.
The labeling of inner vertices ofci′s is(49)fc0=2n,fcn/2=2n+1.
The labeling ofci, for 1≤i≤n/2, is(50)fci=2n+2i−1+3,fcn−1=2n+2,fcn−2=2n+4,fcn/2−2=3n−3,fcn/2−1=3n−1,fcn/2+1=3n−2,fcn/2+2=3n−4.
The labeling ofci, for n/2+2≤i≤n−1, is(51)fci=3n−2−2i−n2−1.
The labeling of the inner vertices ofdi′s is(52)fd0=3n,fdn/2−1=4n−1,fdn−1=3n+2,fd1=3n+3.
The labeling of the verticesdi, for 1≤i≤n/2, is(53)fdi=3n+2i−1+3,fdn/2+1=4n−2,fdn/2=3n+1.
The labeling ofdi, for n/2+2≤i≤n−1, is(54)fdi=4n−2−2i−n2−1.
This labeling is explained in Figure5.
The labeling of outer edges ofai and inner vertices of bi, for 1≤i≤n−1,i≠0,i≠n/2, is(55)f∗aibi=fai2+fbi2=2i−1+32+n+2i−1+32=8i2+8i+n2+4in+2n+2.
The labeling of inner vertices ofci and inner edges of bi, for 1≤i≤n−1,i≠0,i=n/2, is(56)f∗cidi=fci2+fdi2=2n+2i−1+32+3n+2i−1+32=13n2+20in+10n+8i2+8i+2,f∗cn/2+1dn/2+1=fcn/2+12+fdn/2+12=3n−22+4n−22=25n2−28n+8.
The labeling of the edgesaibi, for n/2+2≤i≤n−1, is(57)f∗aibi=fai2+fbi2=n−2−2i−n2−12+2n−2−2i−n2−12=13n2−20in+8i2.
The labeling of the edgescidi, for n/2+2≤i≤n−1, is(58)f∗cidi=fci2+fdi2=3n−2−2i−n2−12+4n−2−2i−n2−12=41n2−36in+8i2.
The labeling of the edgesbici+2, for 1≤i≤n/2, is(59)f∗bici+2=fbi2+fci+22n2+1≤i≤n−3=n+2i−1+32+2n+2i+2−1+32=5n2+8i2+12in+22n+24i+26.
The labeling ofcn/2+1bi+2 is given below:(60)f∗cn/2+1bi+2=fcn/22+fbi+22=3n−22+2n−2−2i−n2−12=3n−22+2n−2−2i+2−n2−12=18n2+4i2−36n+16i−12in+20.
The labeling of the edgesbi+2ci, for n/2+1≤i≤n−3, is(61)f∗bi+2ci=fbi+22+fci2=2n−2−2i−n2−12+3n−2−2i−n2−12=3n−2i+42+4n−2i2=25n2−28in−24n+8i2+16i+16.
The labeling of the edgesa0b0 is given below:(62)f∗a0b0=fa02+fb02=02+n2=n2.
The labeling of the edgesan/2bn/2 is given below:(63)f∗an/2bn/2=fan/22+fn22=12+n+12=n2+2n+2.
The labeling of the edgesc0d0 is given below:(64)f∗c0d0=fc02+fd02=2n2+3n2=4n2+9n2=13n2.
The labeling of edgescn/2dn/2 is given below:(65)f∗cn/2dn/2=fcn/22+fdn/22=2n+12+3n+12=13n2+10n+2.
The labeling of vertices, forbn/2−2cn/2, is given below:(66)f∗bn/2−2cn/2=fbn/2−22+fcn/22=2n−32+2n+12=8n2−8n+10.
The labeling of edgesbn/2−2cn/2 is given below:(67)f∗bn/2−1cn/2+1=fbn/2−12+fcn/2+12=2n−12+3n−22=13n2−16n+5.
The labeling of edges and vertices, forbn/2cn/2+2, is given below:(68)f∗bn/2cn/2+2=fbn/22+fcn/2+22=n+12+3n−42=10n2−22n+17.
The labeling of verticesbici+2, for n/2+1≤i≤n−3, is(69)f∗bici+2=fbi2+fci+22=2n−22+3n−2−2i−n2−12=4n2−8n+4+3n−2−2i+2−n2−12=4n2−8n+4+4n−2i−42=20n2−40n−16in+4i2+16i+20.
The labeling of verticesbn−2c0 is given below:(70)f∗bn−2c0=fbn−22+fco2=n+42+2n2=5n2+8n+16.
The labeling of the edgesbn−1c1 is given below:(71)f∗bn−1c1=fbn−12+fc12=n+22+2n+2i−1+32,i=1=n2+4n+4+2n+32=5n2+16n+13.
The labeling of the edgesbi+2ci, for 1≤i≤n/2−3, is(72)f∗bi+2ci=fbi+22+fci2=n+2i−1+32+2n+2i−1+32=n+2i+52+2n+i+22=5n2+5i2+8in+18n+24i+29.
The labeling of the edgesc0b2 is given below:(73)f∗c0b2=fco2+fb22=2n2+n+2i−1+32=2n2+n+21+32=2n2+n+52=5n2+10n+25.
The labeling of the verticescn−1b1 is given below:(74)f∗cn−1b1=fcn−12+fb12=2n+22+n+2i−1+32=4n2+8n+4+n+32=4n2+8n+4+n2+6n+9=5n2+14n+13.
The labeling of the verticescn−2b0 is given below:(75)f∗cn−2b0=fcn−22+fb02=2n+42+n2=4n2+16n+16+n2=5n2+16n+16.
The labeling of the edgesbn/2cn/2−2 is given below:(76)f∗bn/2cn/2−2=fbn/22+fcn/2−22=n+12+3n−32=n2+2n+1+9n2−18n+9=10n2−16n+10.
The labeling of the edgesbn/2+2cn/2 is given below:(77)f∗bn/2+2cn/2=fbn/2+22+fcn/22=2n−42+2n+12=8n2−12n+17.
The labeling of the edgesbn/2cn/2+2 is given here:(78)f∗bn/2cn/2−2=fbn/22+fcn/2−22=n+12+3n−32=n2+2n+1+9n2−18n+9=10n2−16n+10.
The labeling of the edgesbn/2+1cn/2−1 is given below:(79)f∗bn/2+1cn/2−1=fbn/2+12+fcn/2−12=2n−22+3n−12=4n2−8n+4+9n2−6n+1=13n2−14n+5.
The labeling of the edgesa0a1 is given below:(80)f∗a0a12=fa02+fa12=02+2i−1+32,i=1=02+20+32=02+32=9.
The labeling of the edgesaiai+1, for 1≤i≤n/2−2, is(81)f∗aiai+1=fai2+fai+12=2i−1+32+2i+1−1+32=2i+12+2i+32=8i2+16i+10.
The labeling of the edgesan/2−1an/2 is given below:(82)f∗an/2−1an/22=fan/2−12+fan/22=n−12+12=n2−2n+2.
The labeling of edges with vertices having labelsan/2an/2+1 is(83)f∗an/2an/2+1=fan/22+fan/2+12=12+n−22=n2−4n+5.
The labeling of the edgesan−1a0 is here:(84)f∗an−1a0=fan−12+fa02=22+02=4.
The square sum labeling of the edgesaiai+1, for n/2+2≤i≤n−2, is(85)f∗aiai+1=fai2+fai+12,n2+2≤i≤n−2=n−2−2i−n2−12+n−2−2i+1−n2−12=2n−2i2+2n−2i−22=4n2−8in+4i2+4n2−8in−8n+8i+4i2+4=8n2−16in+8i2−8n+8i+4.
The labeling ofd0d1 is the square sum:(86)f∗d0d1=fd02+fd12=3n2+3n+32=9n2+9n2+18n+9=18n2+18n+9.
The labeling of the edges ofd0d1 is the square sum:(87)f∗didi+1=fdi2+fdi+12=3n+2i−1+32+3n+2i+1−1+32=3n+2i+12+3n+2i+32=9n2+12in+18n+4i2+12i+9+9n2+12in+42+4i+6n+1=18n2+24in+24n+8i2+16i+10.
The labeling of the edgesdn/2−1dn/2 is given below:(88)f∗dn/2−1dn/2=fdn/2−12+fdn/22=4n−12+3n+12=16n2−8n+1+9n2+6n+1=25n2−2n+2.
The labeling of the edgesdn/2dn/2+1 is the square sum:(89)f∗dn/2dn/2+1=fdn/22+fdn/2+12=n+12+4n−22=9n2+6n+1+16n2−16n+5=25n2−10n+5.
The labeling of the edgesdn−1d0 is the square sum:(90)f∗dn−1d0=fdn−12+fd02=3n+22+3n2=9n2+6n+6n+4+9n2=18n2+12n+4.
The labeling of the edgesdidi+1 is the square sum:(91)f∗didi+1=fdi2+fdi+12=4n−2−2i−n2−12+4n−2−2i+1−n2−12=5n−2i2+5n−2i−22=25n2−20in+4i2+25n2−20in−20n+4i2+8i+4=50n2−40in+8i2+8i−20n+4.
From all these computations of the weights of the edges, we can easily see that they are all distinct, that is, the induced labeling functionf∗ is injective. This shows that f is a square sum labeling, and therefore, DPn,2 is a square sum graph.
Here, we prove Case2, when n is odd.
So, this graph shows square sum labeling. LetG≅DPn,2 be the double generalized Petersen graph, where 1≤i≤n−1. We shall show that G is square sum graph by defining a square sum labeling on G.Figure 5
Square sum labeling ofDP12,2.Case 4.
Whenn is odd.
For this case, we define a labelingf:VG⟶0,1,2,3…VG−1, which will show that this labeling is a square sum labeling.
The labeling ofa0 is(92)fa0=1.
The labeling ofb0 is(93)fb0=n+2.
The labeling ofc0 is(94)fc0=n+1.
The labeling ofd0 is(95)fd0=0.
The labeling ofai, for 1≤i≤n−1, is(96)fai=i+1.
The labeling ofbi, for 1≤i≤n−1, is(97)fbi=n+2+i.
The labeling ofci, for 2≤i≤n−1, is(98)fci=3n+i.
The labeling ofdi, for 1≤i≤n−1, is(99)fdi=2n+i+1.
The labeling in this case is explained in Figure6.
The square sum labeling of the edgea0b0 is(100)f∗a0b0=fa02+fb02=12+n+22=n2+4n+5.
The labeling of edges with vertices having labelsaibi, for 1≤i≤n−1, is(101)f∗aibi=fai2+fbi2=i+12+n+i+22=n2+2i2+6i+4n+2in+5.
The square sum labeling of the labelsc0d0 is(102)f∗c0d0=fc02+fd02=n+12+02=n2+2n+1.
The square sum labeling of the labelsc1di is(103)f∗c1di=fc12+fdi2=3n+12+2n+i+12=13n2+10n+i2+4in+2i+2.
The square sum labeling of the labelscidi, for 2≤i≤n−1, is(104)f∗cidi=fci2+fdi2=3n+i2+2n+i+12=13n2+10in+2i2+4n+2i+1.
The square sum labeling ofdid0, for i=n−1, is(105)f∗did0=fdi2+fd02=2n+i+12+02=2n+n−1+12+02=3n2=9.
The square sum labeling ofaia0, for i=n−1, is(106)f∗aia0=fai2+fa02=i+12+12=n−1+12+12=n2+12=n2+1.
The labeling of the edgesaiai+1, for 0≤i≤n−2, is given below:(107)f∗aiai+1=fai2+fai+12=i+12+i+1+12=i+12+i+22=2i2+6i+5.
The labeling of the edgesdidi+1, for 0≤i≤n−2, is given below:(108)f∗didi+1=fdi2+fdi+12=2n+i+12+2n+i+1+12=8n2+8in+12n+2i2+6i+5.
The square sum labeling of the edgesc0bi, for i=n−2, is(109)f∗c0bi=fc02+fbi2=n+12+n+i+22=n+12+n+2+n−22=n+12+n+2+n−22=n+12+2n2=5n2+2n+1.
The square sum labeling of the edgesb0ci, for i=n−2, is(110)f∗b0ci=fb02+fci2=n+22+3n+i2=n+22+3n+n−22=n2+4n+4+16n2+4−16n=17n2−12n+8.
The labeling of the edgesbici is(111)f∗=bici=fbi2+fci2=n+2+i2+3n+i2=n+2+12+3n+i2;bi=n+2+i⇒i=1=n+32+3n+n−12;ci=3n+i⇒i=n−1=n+32+4n−12=17n2−2n+10.
The square sum labeling of the edgesc0bi, for i=2, is(112)f∗c0bi=fc02+fbi2=n+12+n+i+22=2n2+2in+6n+i2+4i+5.
The square sum labeling of the edgescibi+2, for 1≤i≤n−3, is(113)f∗cibi+2=fci2+fbi+22=3n+i2+n+2+i2=10n2+8n+8in+2i2+8i+16.
The square sum labeling of the edgescibi, for 0≤i≤n−3, is(114)f∗cibi=fci2+fbi2=3n+i2+n+2+i2;ci⇒i=i+2=3n+i+22+n+2+i2=10n2+8in+16n+2i2+8i+8.
The square sum labeling of the edgescibi, for 0≤i≤n−3, is(115)f∗cibi=fci2+fbi2=3n+i2+n+2+i2;ci⇒i=i+2=3n+i+22+n+2+i2=10n2+8in+16n+2i2+8i+8.
The labeling of the edgesc1bi is given below:(116)f∗c1bi=fc12+fbi2=3n+12+n+2+i2;ci⇒i=n−1=3n+12+n+2+n−12=3n+12+2n+12=13n2+10n+2.
From all these computations of the weights of the edges, we can easily see that they are all distinct, that is, the induced labeling functionf∗ is injective. This shows that f is a square sum labeling, and therefore, DPn,2 is a square sum graph.Figure 6
Square sum labeling of double generalized Petersen graph [11,2].


## 3. Conclusion

In this study, we have studied the square sum labeling of Generalized Petersen graph and double Generalized Petersen graph, and we prove that these graphs are square sum graphs.



---

*Source: 1872695-2022-02-18.xml*