# A Study onC-Exponential Mean Labeling of Graphs

Authors: Thamaraiselvi Baskaran; Rajasekaran Ganapathy
Journal: Journal of Mathematics
 (2022)

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## Abstract

A functionh is mentioned as a C-exponential mean labeling of a graph GV,E that has s vertices and r edges if h:VG⟶1,2,3,⋯,r+1 is injective and the generated function h∗:EG⟶2,3,4,⋯,r+1 defined by h∗ab=1/ehbhb/haha1/hb−ha, for all ab∈EG, is bijective. A graph which recognizes a C-exponential mean labeling is defined asC-exponential mean graph. In the following study, we have studied the exponential meanness of the path, the graph triangular tree of Tn, CmPn, cartesian product of two paths Pm▫Pn, one-sided step graph of STn, double-sided step graph of 2ST2n, one-sided arrow graph of Ars, double-sided arrow graph of DArs, and subdivision of ladder graph SLt.


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## Body

## 1. Introduction

In the field of mathematics, along with some areas of sciences, graph theory has become an interesting topic of study. A graph labeling is considered as an integer’s assignment to the edges or vertices, or vice versa, subjected to particular conditions. Many mathematicians and scientists have contributed and introduced different kinds of labeling [1–6].In the present study, the graphs considered here are undirected, simple, and finite graphsG=V,E that haves vertices and r edges. Referring to the graph labeling introduced by Gallain, a detailed survey is conducted on graph labeling [4]. Somasundaram and Ponraj [7] originated the theory of mean labeling of graphs. Many mathematicians introduced different aspects of mean labeling. The study of Kannan et al. on the exponential mean labeling of a few different graphs studied through duplicate operations is examined for the present study [8]. Barrientos’ study on alpha graphs has demonstrated the presence of α-labeling of a tree using various vertices and lengths of base path and proved that these trees can be utilized to demonstrate unicycle graphs with α-labeling [9]. Studies on cordial labeling between paths and cycles for a Cartesian product have demonstrated that these Cartesian products, under any conditions, are always cordial and even proved that two path Cartesian products are always cordial [10].Sumathi and Rathi introduced the quotient labeling number for a wide family of ladder graphs, namely, closed triangular ladder, open triangular ladder, closed ladder, open ladder, step ladder, slanting ladder, and open diagonal ladder [11]. Baskar, referring to the flooring function edge labels, defined the logarithmic mean labeling on graphs and studied the logarithmic meanness of different ladder-related graphs [12].Traditionally, the logarithmic mean of any two positive integers is not necessary to be an integer. And, if the logarithmic mean is considered an integer, the flooring of ceiling function is used. The edge label is set through flooring or a ceiling function, which is defined to be the logarithmic mean labeling of graphs. Baskar defined logarithmic mean labeling on graphs by setting the edge labels from flooring function [12]. A graph is considered a logarithmic mean graph if it recognizes logarithmic mean labeling. In 1967, Rosa proposed graceful labeling, known as β-valuation [13], and later, Golomb represented it as graceful labeling [1]. Kaneria et al., in 2010, introduced arrow graph Ank and double arrow graph DAnk[14]. And, in 2015, step grid Stn graph and double step grid graph DStnwere introduced [15]. These graphs were defined to be graceful graphs.Motivated by such works, in this study, we aimed to work to introduce a new class ofC-exponential mean labeling for different ladder graphs, looking at the ceiling function. A graph which recognizes a C-exponential mean labeling is defined as C-exponential mean graph. In the present study, we have examined the exponential meanness of the path, the graph triangular tree of Tn, CmPn, cartesian product of two paths Pm▫Pn, one-sided step graph of STn, double-sided steps graph of 2ST2n, one-sided arrow graph of Ars, double-sided arrow graph of DArs, and subdivision of ladder graph SLt.A functionh is mentioned as a C-exponential mean labeling of a graph GV,Ethat possess s vertices and r edges if h∗:VG⟶1,2,3,⋯,r+1 is injective and the generated function h∗:EG⟶2,3,4,⋯,r+1 defined by
(1)h∗ab=1ehbhbhaha1/hb−ha,for all ab∈EG, is bijective.


### 1.1. Preliminaries

The below-mentioned definitions are essential for the present study.Definition 1.
Letv1,v2,⋯,vn be the consecutive vertices of Pn; a triangular tree is calculated by amalgamating each vi with a leaf (or vertex of degree 1) of Pi. We denote this tree by Tn and refer to Pn as the base of Tn. Note that Tn has size nn+1/2, which means that its order is a triangular number. We say that the first vertex of Pi leaf is amalgamated with the vertex of Pn.Definition 2.
LetPn be a path on n vertices represented by u1,1,u1,2,u1,3,⋯,u1,n and with n−1 edges signified by e1,e2,e3,⋯,en−1, where eirepresents the edge connectingu1,i andu1,i+1, the vertices. On every edgeei, erect a ladder that hasn−n−1 steps counting the edgeei, fori=1,2,3,⋯,n−1. The graph hence drawn is defined as a one-sided step graph, and it is represented bySTn.Definition 3.
LetP2n be a path on 2n vertices w1,1,w1,2,w1,3,⋯,w1,2n with 2n−1 edges u1,u2,u3,⋯,u2n−1, where uirepresents the edge connectingw1,i and w1,i+1, the vertices; on every edge ui, we erect a ladder that hasi+1 steps counting the edge ui, for i=1,2,3,⋯,n, and on everyui, erect a ladder that has2n+1−i steps countingui, for i=n+1,n+2,⋯,2n−1. The graph hence drawn is defined as double-sided step graph, and it is represented by2ST2n.Definition 4.
Anarrow graphArs with breadth s and length r is acquired by joining a vertex w with superior vertices of Ps×Pr by r new edges from one end.Note.
In the graph,Ps×Pr (grid graph on mn vertices) vertices v1,1,v2,1,v3,1,⋯,vm,1 and vertices v1,n,v2,n,v3,n,⋯,vm,n are known as superior vertices (Figure 1) from both ends.Definition 5.
Adouble arrow graph DArs with breadth s and length r is calculated by joining a vertex w with superior vertices of Ps×Pr by s+r new edges from both ends.Definition 6.
A graph, which can be formed from an identified graph GG by dividing up each edge into exactly two segments by positioning intermediate vertices between its two ends, is called a subdivision graph. It is represented bySG.Figure 1
Representation of superior vertex graph.


## 1.1. Preliminaries

The below-mentioned definitions are essential for the present study.Definition 1.
Letv1,v2,⋯,vn be the consecutive vertices of Pn; a triangular tree is calculated by amalgamating each vi with a leaf (or vertex of degree 1) of Pi. We denote this tree by Tn and refer to Pn as the base of Tn. Note that Tn has size nn+1/2, which means that its order is a triangular number. We say that the first vertex of Pi leaf is amalgamated with the vertex of Pn.Definition 2.
LetPn be a path on n vertices represented by u1,1,u1,2,u1,3,⋯,u1,n and with n−1 edges signified by e1,e2,e3,⋯,en−1, where eirepresents the edge connectingu1,i andu1,i+1, the vertices. On every edgeei, erect a ladder that hasn−n−1 steps counting the edgeei, fori=1,2,3,⋯,n−1. The graph hence drawn is defined as a one-sided step graph, and it is represented bySTn.Definition 3.
LetP2n be a path on 2n vertices w1,1,w1,2,w1,3,⋯,w1,2n with 2n−1 edges u1,u2,u3,⋯,u2n−1, where uirepresents the edge connectingw1,i and w1,i+1, the vertices; on every edge ui, we erect a ladder that hasi+1 steps counting the edge ui, for i=1,2,3,⋯,n, and on everyui, erect a ladder that has2n+1−i steps countingui, for i=n+1,n+2,⋯,2n−1. The graph hence drawn is defined as double-sided step graph, and it is represented by2ST2n.Definition 4.
Anarrow graphArs with breadth s and length r is acquired by joining a vertex w with superior vertices of Ps×Pr by r new edges from one end.Note.
In the graph,Ps×Pr (grid graph on mn vertices) vertices v1,1,v2,1,v3,1,⋯,vm,1 and vertices v1,n,v2,n,v3,n,⋯,vm,n are known as superior vertices (Figure 1) from both ends.Definition 5.
Adouble arrow graph DArs with breadth s and length r is calculated by joining a vertex w with superior vertices of Ps×Pr by s+r new edges from both ends.Definition 6.
A graph, which can be formed from an identified graph GG by dividing up each edge into exactly two segments by positioning intermediate vertices between its two ends, is called a subdivision graph. It is represented bySG.Figure 1
Representation of superior vertex graph.


## 2. Main Results

Theorem 8.
Each triangular treeTn is a C-exponential mean graph, for n≥1.Proof.
Assumev1,v2,⋯,vn denote the vertices of the path Pn and each vertex adjoining the path that is represented byvij, for i=1,2,⋯,n,j<i.
Definef:VTn⟶1,2,⋯,nn+1/2 as follows:
(2)fvi=ii+12,ifi=odd,i2−i+22ifi=even,fvij=i2+i−22−j−1,ifi=odd,j<i,i2−i+42+j=1,ifi=even,j<i.
Then, the generated edge labeling is calculated as follows:
Definef∗:ETn⟶2,3,4,⋯,nn+1/2 as follows:
(3)f∗vi,vi+1=vii+1+1,ifi=even,vi+1,ifi=odd,f∗vi,vij=vi+1,ifi=even,j<i,vi−j−1,ifi=odd,i<j.Hence,f is a C-exponential mean labeling of the triangular tree graphTn, for n≥1. A typical example is illustrated in Figure 2.Theorem 9.
The graphCmPn is a C-exponential mean graph, form≥3, n≥1.Proof.
Assumev1,j,v2,j,v3,j⋯,vm,j denote the vertices of the cycle Cn forj=1,2,⋯,n+1. Then, path Pn extended from the cycle through the vertices vi,1,vi,2,vi,3⋯,vi,n+1 for i=1,2,⋯,m.
Definef:VCmPn⟶1,2,⋯,mn+1+1 as follows:
(4)fvi,vj=n+1i−j−1,ifiisodd,andi≤m−12,n+1i+1−j+1,ifiisodd,andi>m−12,ni−1i+i+j−1,ifiiseven,andi<m2,ni−1+i+1+j−1,ifiiseven,andi≥m2.
Hence, the generated edge labeling is calculated as follows:
Definef∗:ECmPn⟶2,3,4,⋯,mn+1+1 as follows:
(5)f∗vi+1,vi+1,1=i+1i+1,ifi≤m−12,n+1i+2,ifi>m−12andm−12<i≤m−1,f∗vm,1,v1,1=m+12n+1+1,when j=1,2,⋯,n.
(6)f∗vi,j,vij+1=ni+1+i+j,ifiiseven,andi≤m−12,ni−1+i+j+1,ifiiseven,andi>m−12,n+1i−j−1,ifiisodd,andi≤m2,n+1i−j+2,ifiiseven,andi>m2.
Hence,f is a C-exponential mean labeling of the graphCnPn, for n≥3. Figure 3 depicts an example of the aforementioned labeling.Theorem 10.
The graphPm▫Pn is a C-exponential mean graph, for m,n≥1Proof.
The Cartesian product of the graphsG and H is the graph G▫Hthat has vertex set VG and edge setEG. In this article, the vertices of the graph Pm▫Pn are presented as a matrix with n rows and m columns. Moreover, we denote vi,j as the vertex which lies at the i-th row and j-th column where 1≤i≤m and 1≤j≤n.
Definef:VPm▫Pn⟶1,2,⋯,m2n−1−n+1 as follows:
(7)fvi,vj=2mj−1−j−2+i−1,for1≤i≤mand1≤j≤n.
Then, the generated edge labeling is calculated as follows:
Definef∗:EPm▫Pn⟶2,3,4,⋯,m2n−1−n+1 as follows:
(8)f∗vij,vij+1=2mj−1−j−3+i−1,for1≤i≤mand1≤j≤n.fvij,vij+1=m2j−1−j−2+i−1,for1≤i≤mand1≤j≤n.
Hence,f is a C-exponential mean labeling of the graphPm▫Pn, for n≥2. Figure 4 illustrates a representative example of the labeling described above.Theorem 11.
The graphSTn is a C-exponential mean graph, for n≥2.Proof.
Assumev1,1,v1,2,v1,3,⋯⋯v1,n,v2,1,v2,2,v2,3,⋯⋯v2,n,v3,1,v3,2,v3,3,⋯⋯v3,n−1,v4,1,v4,2,v4,3,⋯⋯v4,n−2,⋯.vn,1,vn,2denote the vertices of the step graphSTn.
LetG=STn be the step ladder graph with n−i−1 steps for 1≤i≤n.
Letv1,j be the n vertices on the base where 1≤j≤n.
Letv2,j be the n vertices on the second stage above the base for 1≤j≤n.
Letv3,j be the n−1 vertices on the third step for 1≤j≤n−1.
Proceeding like this, we have vertices forn−i−1 steps.
Now the vertices ofSTn is denoted byvi,j.
Invi,j, i signifies the row (calculated through bottom to top) and j signifies the column (calculated through left to right) in which the vertex occurs.
Now, the graphSTn of vertices and edges are n2+n−1 with degv1,1=degv1,n=degv2,n=degvn,1=2; degvi,n−i+2=2 for 3≤i≤n; degvi,1=3 for 2≤i≤n−1; degv1,j=3 for 2≤j≤n−1; and degv1,j=4 for 1≤i≤n−1,1≤j≤n−1, and j≠n−i+2.
But,G,SG=2 and ΔG=4.
Definef:VSTn⟶1,2,⋯,n2+n−1 as follows:
(9)fvi,1=n−i−1,for1≤i≤n,fvij1=3n−3j+5+2∑j≥3nn−j+3−i−1,for1≤i≤nand2≤j≤n.
Hence, the generated edge labeling is calculated as follows:
Definef∗:ESTn⟶2,3,4,⋯,n2+n−1 as follows:
(10)f∗vi,1,vi+1,1=n−i−1,for1≤i≤n,f∗vi,j,vi+1,j=3n−3j+5+2∑j≥3nn−j+3−i−1,for1≤i≤n−1and2≤j≤n.
Hence,f is a C-exponential mean labeling of the graph STn. Thus, the graph STn is a C-exponential graph for n≥2. A characteristic example of the labeling mentioned above is shown in Figure 5.Theorem 12.
The graph2ST2n is a C-exponential mean graph, for n≥2.Proof.
Assumew1,1,w1,2,w1,3,⋯,w1,n,w2,n,w3,n,⋯,w2,2n,w3,1,w3,2,w3,3,⋯,w3,2n−2,⋯.,w4,1,w4,2,w4,3,⋯,w4,2n−4,⋯..,wn+1,1,wn+1,2 denote the vertices of the double-sided step graph2ST2n. Inwi,j, i signifies the row (calculated through bottom to top) and j signifies the column (calculated through left to right) in which the vertex occurs.
Definel:V2ST2n⟶1,2,⋯,n2n+3 as follows:
(11)lw1,j=2,j=1,wi,j−1+n+2j−3,1<j≤n,wi,j−1+4n+3−2j,1<j≤n,lw2,j=wi−1,j−i−1,for1≤j≤2n,2<i≤n+1and1≤j≤2n+4−2i,lwi,j=wi−1,j+1−1.
The above-defined labeling pattern gives rise tol as an injective map and defines l∗:E2ST2n⟶2,3,⋯,n2n+3 as follows:
(12)f∗uv=1efvfvfufu1/fv−fu,for all uv∈E2ST2n, is defined as bijective.
(13)l∗w1,j,wi,j+1=wi,j+j+1,1≤j≤n,wi,j+2n−j+1,n+1≤j≤2n,l∗w2,j,wi,j+1=wi,jw1,j+1−1,for1≤j<2n,wi,j+2n−j+1,n+1≤j≤2n,fori>2,1≤j<2n+4−2i,l∗wi,jwi,j+1=wi−1,j+1wi−1,j+2−1,l∗w1,j,w2,j=w1,j,for1≤j≤2n,wi,j+2n−j+1,n+1≤j<2n,fori>1,1≤i<n,2≤j≤2n+4−2i,l∗wi,jwi+1,j−1=wi,j.
Hence,l is a C- exponential mean labeling of2ST2n, Hence, the graph, for 2ST2n, is a C-exponential mean graph for n≥2. Figure 6 displays a distinctive illustration of the labeling stated before.Theorem 13.
The graphArs is a C-exponential mean graph, where r≥2 and s≥2.Proof.
AssumeH=Ars is an arrow graph calculated by connecting a vertex w with superior vertices of Ps×Pr by two new edges.
Letxi,ji=1,2;j=1,2,⋯,m be vertices of Ps×Pr.
Joinw with xi,1i=1,2 by 2 new edges to obtain H.VH=2m+1 and EH=3m.
ξ:VArs⟶1,2,⋯,2rs−r−s+3 by using C-exponential mean labeling formula for all uv∈EH is defined as bijective.
(14)ξw=1,ξxi,j=3+i−1+2s−1j−1,i=1,2,⋯,sandj=1,2,⋯,r−1.
The above-defined labeling pattern gives rise toξ as an injective map and defines ξ∗:EArs⟶2,3,⋯,2rs−r−s+3 as follows:
(15)ξ∗w,x1,1=2,ξ∗w,x2,1=3,ξ∗xi,j,xi+1,j=3+i+2s−1j−1,i=1,2,⋯,sandj=1,2,⋯,r,ξ∗xi,j,xi,j+1=3−s+j2s−1+i,i=1,2,⋯,sandj=1,2,⋯,r−1.
Hence, the graphArs is a C-exponential mean graph, for r≥2 and s≥2. A sample example of the previously mentioned labeling is shown in Figure 7.Theorem 14.
The graphDArs is a C-exponential mean graph, where r≥2 and s≥2.Proof.
AssumeH=DArs is a double arrow graph calculated by joining two vertices w,y with Ps×Pr by four new edges on both sides.
Letxi,ji=1,2,⋯,s;j=1,2,⋯,r be vertices of Ps×Pr. Join w with xi,1i=1,2,⋯,s and y with xi,ri=1,2,⋯,s by four new edges to obtainH. VG=2rs−r−s+5 and EG=2rs−r−s+5.
Defineξ:VDArs⟶1,2,⋯,2rs−r−s+5 as follows:
(16)ξw=1,ξy=2rs−r−s+5,ξxi,j=3+i−1+2s−1j−1,i=1,2,⋯,sandj=1,2,⋯,r−1,ξx1,r=2rs−r−2s+4,ξxi,r=2rs−r−2s+4+i,i=2,3,⋯,s.
Then, the generated edge labeling is calculated as follows:
ξ∗:EDArs⟶2,3,4,⋯,2rs−r−s+5by using C-exponential mean labeling formula, for all uv∈EH, is defined as bijective.
(17)ξ∗w,x1,1=2,ξ∗=w,xs,1=3,ξ∗xi,j,xi+1,j=3+i+2s−1j−1,i=1,2,⋯,s−1;j=1,2,⋯,r−1,ξ∗x1,r,x2,r=2rs−r−2s+5,ξ∗xi,r,xi+1,r=2rs−r−2s+7+i−2,i=2,3,⋯,r,ξ∗xi,j,xi,j+1=3−s+j2s−1+i,i=1,2,⋯,s;j=1,2,⋯,r−1,ξ∗x1,r,y=ξx1,r+2,ξ∗xs,r,y=ξxs,r+1.
Hence, the graphDArs is a C-exponential mean graph, for r≥2. Figure 8 displays an illustration of the labeling from earlier as an example.Theorem 15.
The subdivision of ladder graphSLt is a C-exponential mean graph, for t≥2.Proof.
AssumeH=Lt. The ladder graph Lt is defined as Lt=Pt×K2, where Pt is a path with × signifing the cartesian product. Let r1,r2,⋯,rt,s1,s2,s3,⋯,st be the ladder vertices Lt. Let si′ be the lately added vertex joining si andsi+1, ri′ be the newly added vertex between ri and ri+1 and qi be the lately added vertex joiningri and si. Clearly, G=SLt has 5t−2 vertices and 6t−4 edges.
Defineψ:VSLt⟶1,2,⋯,6t−4+1 as follows:
(18)ψr1=1,ψr1=6i−1i=oddand1<i<t,23i−2i=evenand1≤i≤t,ψs1=3,ψr1=6i−1i=evenand1≤i≤t,23i−2i=oddand1<i≤t,ψq1=2,ψqi=6i−5,1<i≤t,ψri′=32i+1,i=oddand1≤i<t,6i−1,i=evenand1≤i<t,ψsi′=6i−1,i=oddand1≤i<t,32i+1,i=evenand1≤i<t.
Hence, the generated edge labeling is calculated as follows:
Defineψ∗:ESLt⟶2,3,⋯,6t−4+1 as follows, for 1≤i<t.
(19)ψri,ri′=6i−1,i=odd,6i−2,i=even,ψri′,ri+1=6i,i=even,32i+1,i=odd,ψsi,si′=23i−1,i=odd,6i−1,i=even,ψ∗si′,si+1=maxψsi′,ψsi+1,ψ∗ri,qi=maxψri,ψqifori≤i≤t,ψ∗qi,si=maxψqi,ψsifori≤i≤t.
Hence,ψ is a C-exponential mean graphSLt, for t≥2. A representation of the prior labeling is shown as an example in Figure 9.Figure 2
AC-exponential mean labeling of T7Figure 3
AC-exponential mean labeling of CnPnFigure 4
AC-exponential mean labeling of Pm▫PnFigure 5
AC-exponential mean labeling of ST7Figure 6
AC-exponential mean labeling of 2ST8Figure 7
AC-exponential mean labeling of A62Figure 8
AC-exponential mean labeling of DA52Figure 9
SL5 is a C-exponential mean labeling


## 3. Conclusion

TheC-exponential mean labeling of tree, cycle-path, step graph, ladder graphs, arrow graphs, and subdivision of ladder graph was introduced and discussed in this work using graph operations.



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