# Assignment Computations Based onCexp Average in Various Ladder Graphs

Authors: A.Rajesh Kannan; P. Manivannan; K. Loganathan; K. Prabu; Sonam Gyeltshen
Journal: Journal of Mathematics
 (2022)

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## Abstract

This study introduces theCexp average assignments and investigates its properties using various ladder graphs. The ladder graphs can be found in every communication networks. Ladder networks are increasingly being used in everyday life for monitoring and environmental applications such as domestic, military, surveillance, industrial, medical applications, and traffic management. These datasets are afflicted by the average representation of the graph structure. It aids in the visualisation and comprehension of data analysis. The Cexp labeling is used in sensor networks, adhoc networks, and other applications. It also efficiently creates a communication network after using noise reduction methods to remove salt and pepper noise.


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## Body

## 1. Introduction

A graph labeling is the assignment of labels, conventionally indicated by integers, to edges and/or vertices of a graph in the mathematical domain of graph theory. The concept of labeling may be applied to many areas of graph theory, for example, in automata theory and formal language theory. We use [1–5] for notations and nomenclature. We recommend [6] for a thorough examination of graph labeling. Let Pn be a path on n nodes denoted by u1,μ, where 1≤μ≤n, and with n−1 lines denoted by e1,δ, where 1≤δ≤n−1, where eμ is the line joining the vertices u1,μ and u1,μ+1. On each edge eδ, erect a ladder with n−μ−1 steps including the edge eμ, for μ=1,2,3,…,n−1. The resulting graph is called the one-sided step graph, and it is denoted by STn. Let G1 and G2 be any two graphs with p1 and p2 vertices, respectively. Then, G1×G2 is the Cartesian product of two graphs. A ladder graph Ln is the graph P2×Pn. The graph G°Sm is obtained from G by attaching m pendant vertices to each vertex of G. The triangular ladder TLn, for n≥2, is a graph obtained from two paths by u1,u2,…un and v1,v2,…vn by adding the edges uμvμ,1≤μ≤n and uμvμ+1,1≤μ≤n−1. The slanting ladder SLn is a graph obtained from two paths u1,u2,…un and v1,v2,…vn by joining each vμ, with uμ+1,1≤μ≤n−1. The graph Dn∗ having the vertices aμ,δ:1≤μ≤n,δ=1,2,3,4 and its edge set is aμ,1aμ+1,1,aμ,3aμ+1,3:1≤μ≤n−1∪aμ,1aμ,2,aμ,2aμ,3,aμ,3aμ,4,aμ,4aμ,1:1≤μ≤n.


## 2. Literature Survey

In [7], the authors talked about the F-root square mean labeling for line graph of the path, cycle, star, Pn°S1,Pn°S2,Pn;S1,SPn°S1, ladder, slanting ladder, the crown graph Cn°S1, and the arbitrary subdivision of S3. The authors in [8] discussed 1,1,0F-face mean labeling some planar graphs and, in [9], face labelings of type 1,1,1 for generalized prism. Alanazi et al. explained the classical meanness of the graphs, the one-sided step graph STn, double-sided step graph 2ST2n, planar grid Pm×Pn, ladder graph Ln, graph Ln°Sm for m≤2, triangular ladder graph TLn, graph TLn°Sm for m≤2, graph SLn°Sm for m≤2, slanting ladder graph SLn, graph SLn°Sm for m≤2, graph Dn∗, diamond ladder graph Dln, and latitude ladder graph LLn in [10–12]. Dafik slamin et al. highlighted the super a,d-edge-antimagic total properties of triangular book and diamond ladder graphs in [13]. Moussa and Badr discussed the odd gracefulness of few ladder graphs and proved that ladder and subdivision of ladder graphs with pendent edges are odd graceful in [14]. In [15], the authors emphasized the significance of exponential mean labeling of graphs, and they examined the exponential mean labeling of some graphs obtained from duplicating operations. Inspired by such tremendous works of researchers in the region of graph assignments in [16–25], we defined Cexp average assignment of graphs. A Cexp average of two integers is not always an integer. Consequently, Cexp average assignment must be an integer; we may get ceiling function by considering the integral part. In this study, our conversation and attempt is to examine the various assignment techniques on Cexp average assignment for few ladder graphs.


## 3. Methodology

A functionΨ is known as an Cexp average assignment of G if Ψ:VG⟶N−q+2,q+3,…,∞ is one to one and the instigated bijective function Ψ∗:EG⟶N−1,q+2,q+3,…,∞ characterized by(1)Ψ∗uv=1eXvXu1/Y,where Xw=ΨwΨw, Y=Ψv−Ψu, q is the number of edges, and N is the set of all natural numbers. A graph that concedes an Cexp average assignment is known as a Cexp average assignment graph.Figure1 shows the Cexp average assignment of the graph SL2°S1.Figure 1
AnCexp average assignment of the graph, SL2°S1.


## 4. Main Results

Theorem 1.
The one-sided step graphSTn is an Cexp average assignment graph, for n≥2.Proof.
Make the vertex assignmentΨ:VSTn⟶N−n2+n,n2+n+1,…,∞, Ψu1,μ=n2−1+μ,for2≤μ≤n,andΨuλ,μ=1+n−λ2+μ−1,for2≤λ≤nand1≤μ≤n+2−λ.
Consequently, the instigated edge assignmentΨ∗ is acquired as follows:(2)Ψ∗uλ,μuλ+1,μ=−2nλ+n−λ+λ2+n2+μ,forn−1≥λ≥1and1−λ+n≥μ≥1,Ψ∗u1,μu1,μ+1=n2+μ,for1≤μ≤n−1,andΨ∗uλ,μuλ,μ+1=1+n−λ2+μ,for2≤λ≤nand1≤μ≤n+1−λ.
As a result, forn≥2, Ψ is an Cexp average assignment and the one-sided step graph STn is an Cexp average assignment graph.Theorem 2.
The graphPm×Pn is an Cexp average assignment graph, for m≤4 and n≥2.Proof.
Case (i):m=2.Make the vertex assignment,Ψ:VP2×Pn⟶N−3n,3n+1,…,∞.Ψvλμ=λ−3+3μ,for1≤λ≤2and1≤μ≤n.Consequently, the instigated edge assignmentΨ∗ is acquired as follows.Ψ∗vλμvλμ+1=λ−1+3μ,for1≤λ≤2and1≤μ≤n−1.Ψ∗v1μv2μ=−13μ,for1≤μ≤n.Case (ii):m=3.Make the vertex assignment,Ψ:VP3×Pn⟶N−5n−1,5n+1,…,∞.Ψvλμ=λ−5+5μ,for1≤λ≤3and2≤μ≤n.Ψvλ1=λ,1≤λ≤2,4,λ=3.Consequently, the instigated edge assignmentΨ∗ is acquired as follows:Ψ∗vλμvλμ+1=λ−2+5μ,for1≤λ≤3and2≤μ≤n−1Ψ∗vλμvλ+1μ=λ−4+5μ,for1≤λ≤2and2≤μ≤nΨ∗vλ1vλ2=λ+3,for1≤λ≤3,Ψ∗vλ1vλ+11=1+λ,for1≤λ≤2,Case (iii):m=4 and n≥3.Make the vertex assignment:Ψ:VP4×Pn⟶N−7n−2,7n−1,…,∞Ψvλμ=λ−7+7μ,for1≤λ≤4and3≤μ≤nΨvλ2=λ+7,for1≤λ≤4Ψvλ1=λ,1≤λ≤2λ+1,3≤λ≤4
Consequently, the instigated edge assignmentΨ∗ is acquired as follows:(3)Ψ∗vλμvλ+1μ=λ−6+7μ,for1≤λ≤3and3≤μ≤n,Ψ∗vλ2vλ+12=8+λ,for1≤λ≤3,Ψ∗vλ1vλ+11=λ+1,1≤λ≤2,5,λ=3,Ψ∗vλμvλμ+1=λ−3+7μfor2≤λ≤4and1≤μ≤n−1,andΨ∗v1μv1μ+1=8μ−4,1≤μ≤2,7μ−2,3≤μ≤n−1.
As a result,Ψ is an Cexp average assignment and the graph Pm×Pn is an Cexp average assignment graph, for m≤4.Corollary 1.
Every Ladder graphLn=P2×Pn is an Cexp average assignment graph.Theorem 3.
The graphLn°Sm is an Cexp average assignment graph, for n≥2 and m≤2.Proof.
Case (i):m=1.Make the vertex assignment:Ψ:VLn°S1⟶N−5n,5n+1,…,∞,(4)Ψx1λ=2,λ=1,−1+5λ,2≤λ≤n,Ψu1λ=−4+5λ,for1≤λ≤n,Ψvλ=4,λ=1,−2+5λ,2≤λ≤nand,Ψuλ=3,λ=1,−3+5λ,2≤λ≤n.Consequently, the instigated edge assignmentΨ∗ is acquired as follows:(5)Ψ∗vλx1λ=3,λ=1,−1+5λ,2≤λ≤n.Ψ∗uλw1λ=5λ−3,for1≤λ≤n,Ψ∗uλvλ=4,λ=1,−2+5λ,2≤λ≤n,Ψ∗vλvλ+1=1+5λ,for1≤λ≤n−1andΨ∗uλuλ+1=5λ,for1≤λ≤n−1.Case (ii):m=2.Make the vertex assignment:Ψ:VLn°S2⟶N−7n,7n+1,…,∞,(6)Ψx2λ=8,λ=1,−5+7λ,2≤λ≤n,λ=2k,k∈ℕ,−2+7λ,2≤λ≤n,λ=2k+1,k∈ℕ,Ψx1λ=2λ+3,1≤λ≤2,−6+7λ,3≤λ≤n,λ=2k+2,k∈ℕ,−3+7λ,2≤λ≤n,λ=2k+1k∈ℕ,Ψw2λ=2,λ=1,−1+7λ,2≤λ≤n,λ=2k,k∈ℕ,−4+7λ,2≤λ≤n,λ=2k+1,k∈ℕ,Ψw1λ=1,λ=1,−3+7λ,2≤λ≤n,λ=2k,k∈ℕ,−6+7λ,2≤λ≤n,λ=2k+1,k∈ℕ,Ψvλ=4,λ=1,−4+7λ,2≤λ≤n,λ=2k,k∈ℕ,−1+7λ,2≤λ≤n,λ=2k+1,k∈ℕ,Ψuλ=3,λ=1,−2+7λ,2≤λ≤n,λ=2k,k∈ℕ,−5+7λ,2≤λ≤n,λ=2k+1,k∈ℕ$k∈ℕ..Consequently, the instigated edge assignmentΨ∗ is acquired as follows:(7)Ψ∗vλx2λ=7λ−4,2≤λ≤n,λ=2k,k∈ℕ,−1+7λ,1≤λ≤n,λ=2k+1,k∈ℕ,Ψ∗vλx1λ=7λ−5,1≤λ≤n,λ=2k,k∈ℕ,−2+7λ,4≤λ≤n,λ=2k+1,k∈ℕ,Ψ∗uλw2λ=7λ−1,1≤λ≤n,λ=2k,k∈ℕ,−4+7λ,1≤λ≤n,λ=2k+1,k∈ℕ,Ψ∗uλw1λ=2,λ=1,−2+7λ,2≤λ≤n,λ=2k,k∈ℕ,−5+7λ,1≤λ≤n,λ=2k+1,k∈ℕ,Ψ∗uλvλ=7λ−3,for1≤λ≤n,Ψ∗vλvλ+1=7λ+1,for1≤λ≤n−1,andΨ∗uλuλ+1=8,λ=1,7λ,2≤λ≤n−1.As a result,Ψ is an Cexp average assignment and the graph Ln°Sm is an Cexp average assignment graph, for n≥2 and m≤2.Theorem 4.
The triangular ladder graphTLn is an Cexp average assignment graph, for n≥2.Proof.
Make the vertex assignment;Ψ:VTLn⟶N−4n−1,4n+1,…,∞,(8)Ψvλ=4λ−1,1≤λ≤n−1,4n−2,λ=n,and,Ψuλ=4λ−3,for1≤λ≤n.
Consequently, the instigated edge assignmentΨ∗ is acquired as follows:(9)Ψ∗vλuλ+1=4λ,for1≤λ≤n−1,Ψ∗uλvλ+1=1+4λ,for1≤λ≤n−1,Ψ∗uλvλ=−2+4λ,for1≤λ≤n, andΨ∗uλuλ+1=−1+4λ,for1≤λ≤n−1.
As a result,Ψ is an Cexp average assignment and the triangular ladder graph TLn is an Cexp average assignment graph, for n≥2.Theorem 5.
The graphTLn°Sm is an Cexp average assignment graph, for n≥2 and m≤2.Proof.
Case (i):m=1.Make the vertex assignment;Ψ:VTLn°S1⟶N−6n−1,6n,…,∞,(10)Ψx1λ=3,λ=1,−3+6λ,2≤λ≤n,Ψu1λ=−6+7λ,1≤λ≤2,−5+6λ,3≤λ≤n,Ψvλ=6λ−2,for1≤λ≤n,andΨuλ=−3+5λ,1≤λ≤2,−4+6λ,3≤λ≤n.Consequently, the instigated edge assignmentΨ∗ is acquired as follows:(11)Ψ∗vλx1λ=3,λ=1,−2+6λ,2≤λ≤n.Ψ∗uλw1λ=−4+6i,for1≤λ≤nandΨ∗uλvλ=4,λ=1,−3+6λ,2≤λ≤n,Ψ∗vλuλ+1=6λ,for1≤λ≤n−1,Ψ∗vλvλ+1=1+6λ,for1≤λ≤n−1,andΨ∗uλuλ+1=−1+6λ,for1≤λ≤n−1.Case (ii):m=2.Make the vertex assignment;Ψ:VTLn°S2⟶N−8n−1,8n,…,∞,(12)Ψx2λ=9,λ=1,−6+8λ,2≤λ≤n,Ψx1λ=4,λ=1,−7+8λ,2≤λ≤n,Ψw2λ=3,λ=1,−2+8λ,2≤λ≤n,Ψw1λ=1,λ=1,−4+8λ,2≤λ≤n,Ψvλ=6,λ=1,−5+8λ,2≤λ≤n,andΨuλ=2,λ=1,−3+8λ,2≤λ≤n.Consequently, the instigated edge assignmentΨ∗ is acquired as follows:(13)Ψ∗vλx2λ=8,λ=1,−5+8λ,2≤λ≤n,Ψ∗vλx1λ=5,λ=1,−6+8λ,2≤λ≤nΨ∗uλw2λ=3,λ=1,−2+8λ,2≤λ≤n,Ψ∗uλw1λ=2,λ=1,−3+8λ,2≤λ≤n,Ψ∗vλuλ+1=6,λ=1,8λ,2≤λ≤n−1,Ψ∗uλvλ=8λ−4,for1≤λ≤n,Ψ∗vλvλ+1=9,λ=1,−1+8λ,2≤λ≤n−1,andΨ∗uλuλ+1=7,λ=1,1+8λ,2≤λ≤n−1.
As a result,Ψ is an Cexp average assignment and the graph TLn°Sm is an Cexp average assignment graph, for n≥2 and m≤2.Theorem 6.
The slanting ladder graphSLn is an Cexp average assignment graph, for n≥2.Proof.
Make the vertex assignment;Ψ:VSLn⟶N−3n−1,3n,…,∞,(14)Ψvn=3n−2,Ψvλ=3λ,for1≤λ≤n−1,Ψuλ=3λ−4,for2≤λ≤n,andΨu1=1.
Consequently, the instigated edge assignmentΨ∗ is acquired as follows:(15)Ψ∗vλuλ+1=3λ,for1≤λ≤n−1,Ψ∗vn−1vn=−2+3n,Ψ∗vλvλ+1=3λ+2,for1≤λ≤n−2,andΨ∗uλuλ+1=2,λ=1,3λ−2,2≤λ≤n−1.
As a result,Ψ is an Cexp average assignment and the graph SLn is an Cexp average assignment graph.Theorem 7.
The graphSLn°Sm is an Cexp average assignment graph, for n≥2 and m≤2.Proof.
Casei: m=1 and n≥3.Make the vertex assignment,Ψ:VSLn°S1⟶N−5n−1,5n,…,∞.(16)Ψx1λ=7,λ=1,1+5λ,2≤λ≤n−1,−3+5n,λ=n,Ψw1λ=3λ−2,1≤λ≤2,−7+5λ,3≤λ≤n,Ψvλ=6,λ=1,5λ,2≤λ≤n−1,5n−2,λ=n,andΨuλ=1+λ,1≤λ≤2,−6+5λ,3≤λ≤n.Consequently, the instigated edge assignmentΨ∗ is acquired as follows:(17)Ψ∗vλx1λ=7,λ=1,1+5λ,2≤λ≤n−1,−2+5n,λ=n,Ψ∗uλw1λ=2,λ=1,−6+5λ,2≤λ≤n,Ψ∗vλuλ+1=5λ,for1≤λ≤n−1,Ψ∗vλvλ+1=5λ+3,1≤λ≤n−2,−3+5n,λ=n−1,Ψ∗uλuλ+1=3λ,1≤λ≤2,5λ−3,3≤λ≤n−1.Case (ii):m=2 and n≥3.Make the vertex assignment;Ψ:VSLn°S2⟶N−7n−1,7n,…,∞,(18)Ψx2λ=11,λ=1,1+7λ,2≤λ≤n−3,λ=2k,k∈ℕ,−2+7λ,2≤λ≤n−3,λ=2k+1,k∈ℕ,−9+7n,λ=n−2,n=2k,k∈ℕ,−16+7n,λ=n−2,n=2k+1,k∈ℕ,−6+7n,λ=n−1,and−2+7n,λ=n,Ψx1λ=9,λ=1,7λ,2≤λ≤n−3,λ=2k,k∈ℕ,−3+7λ,2≤λ≤n−3,λ=2k+1,k∈ℕ,−12+7n,λ=n−2,n=2k,k∈ℕ,−17+7n,λ=n−2,n=2k+1,k∈ℕ,−8+7n,λ=n−1,n=2k,k∈ℕ,−7+7n,λ=n−1,n=2k+1,k∈ℕ,−4+7n,λ=n,Ψw2λ=7λ−5,1≤λ≤2,−5+7λ,3≤λ≤n−1,λ=2k+2,k∈ℕ,−8+7λ,3≤λ≤n−1,λ=2k+1,k∈ℕ,−7+7n,λ=n,n=2k,k∈ℕ,−8+7n,λ=n,n=2k+1,k∈ℕ,Ψw1λ=1,λ=1,−5+5λ,2≤λ≤3,−7+7λ,4≤λ≤n−1,λ=2k+2,k∈ℕ,−10+7λ,4≤λ≤n−1,λ=2k+3,k∈ℕ,−11+7n,λ=nandλ=n−1,−10+7n,λ=nandλ=n,Ψvλ=8,λ=1,2+7λ,2≤λ≤n−3,λ=2k,k∈ℕ,−1+7λ,2≤λ≤n−3,λ=2k+1,k∈ℕ,−13+7n,λ=n−2,n=2k,k∈ℕ,−15+7n,λ=n−2,n=2k+1,k∈ℕ,−5+7n,λ=n−1,−3+7n,λ=n,Ψuλ=λ+2,1≤λ≤2,−6+7λ,3≤λ≤n−1,n=2k+2,k∈ℕ,−9+7λ,3≤λ≤n−1,n=2k+1,k∈ℕ,−10+7n,λ=n,k∈ℕ,−9+7n,λ=n,n=2k+1,k∈ℕ.Consequently, the instigated edge assignmentΨ∗ is acquired as follows:(19)Ψ∗vλx2λ=10,λ=1,2+7λ,2≤λ≤n−3,λ=2k,k∈ℕ,−1+7λ,2≤λ≤n−3,λ=2k+1,k∈ℕ,−11+7n,λ=n−2,n=2k,k∈ℕ,−15+7n,λ=n−2,n=2k+1,k∈ℕ,−5+7λ,λ=n−1,−2+7n,λ=n,Ψ∗vλx2λ=10,λ=1,2+7λ,2≤λ≤n−3,λ=2k,k∈ℕ,−1+7λ,2≤λ≤n−3,λ=2k+1,k∈ℕ,−11+7n,λ=n−2,n=2k,k∈ℕ,−15+7n,λ=n−2,n=2k+1,k∈ℕ,−5+7λ,λ=n−1,−2+7n,λ=n,Ψ∗vλx1λ=9,λ=1,1+7λ,2≤λ≤n−3,λ=2k,k∈ℕ,−2+7λ,2≤λ≤n−3,λ=2k+1,k∈ℕ,−12+7n,λ=n−2,n=2k,k∈ℕ,−16+7n,λ=n−2,n=2k+1,k∈ℕ,−6+7n,λ=n−1,−6+7n,λ=n,Ψ∗uλw2λ=4λ−1,1≤λ≤2,−5+7λ,3≤λ≤n−1,λ=2k+2,k∈ℕ,−8+7λ,3≤λ≤n−1,λ=2k+1,k∈ℕ,−8+7n,λ=n,Ψ∗uλw1λ=2,λ=1,−7+6λ,2≤λ≤3,−6+7λ,4≤λ≤n−1,λ=2k+2,k∈ℕ,−9+7λ,4≤λ≤n−1,λ=2k+3,k∈ℕ,−10+7n,λ=n,n=2k,k∈ℕ,−9+7n,λ=n,n=2k,k∈ℕ,Ψ∗vλvλ+1=12,λ=1,7λ+4,2≤λ≤n−3,7n−9,λ=n−2,n=2k,k∈ℕ,7n−10,λ=n−2,n=2k+1,k∈ℕ,7n−4,λ=n−1,Ψ∗vλuλ+1=6,λ=1,7λ,2≤λ≤n−1,Ψ∗uλuλ+1=4λ,1≤λ≤2,7λ−4,3≤λ≤n−2,7n−13,λ=n−1,n=2k,k∈ℕ,7n−11,λ=n−1,n=2k,k∈ℕ.Case (iii):m=1,2 and n=2. An Cexp average assignment of the graphs SL2°S1 and SL2°S2 are shown in Figures 1 and 2.
As a result,Ψ is an Cexp average assignment and the graph SLn°Sm is an Cexp average assignment graph, for n≥2 and m≤2.Figure 2
AnCexp average assignment of the graph SL2°S2.Theorem 8.
The graphDn∗ is an Cexp average assignment graph, for n≥2.Proof.
Make the vertex assignment;Ψ:VDn∗⟶N−6n,6n+1,…,∞,(20)Ψaλ,4=−1+6λ,for1≤λ≤n,Ψaλ,3=−3+6λ,for1≤λ≤n,Ψaλ,2=−5+6λ,for1≤λ≤n,andΨaλ,1=−2+6λ,for1≤λ≤n.
Consequently, the instigated edge assignmentΨ∗ is acquired as follows:(21)Ψ∗aλ,4aλ,1=−1+6λ,for1≤λ≤n,Ψ∗aλ,3aλ,4=−2+6λ,for1≤λ≤n,Ψ∗aλ,2aλ,3=−4+6λ,for1≤λ≤n,Ψ∗aλ,1aλ,2=−3+6λ,for1≤λ≤n,Ψ∗aλ,3aλ+1,3=6λ,for1≤λ≤n−1,andΨ∗aλ,1aλ+1,1=1+6λ,for1≤λ≤n−1.
As a result,Ψ is an Cexp average assignment and the graph Dn∗ is an Cexp average assignment graph, for n≥2.Theorem 9.
The diamond ladder graphDln is an Cexp average assignment graph, for n≥1.Proof.
Make the vertex assignment;Ψ:VDln⟶N−8n−1,8n,…,∞,(22)Ψzλ=1,λ=1,−2+4λ−−1λ+1+12,2≤λ≤2nandλiseven,−2+4λ−−1λ+1+12,3≤λ≤2nandλisodd,Ψyλ=−3+8λ,for1≤λ≤n,andΨxλ=−5+8λ,for1≤λ≤n.
Consequently, the instigated edge assignmentΨ∗ is acquired as follows:(23)Ψ∗yλz2λ=−2+8λ,for1≤λ≤n,Ψ∗yλz2λ−1=−5+8λ,for1≤λ≤n,andΨ∗xλz2λ=−3+8λ,for1≤λ≤n,Ψ∗xλz2λ−1=−6+8λ,for1≤λ≤n,Ψ∗z2λz2λ+1=8λ,for1≤λ≤n−1,Ψ∗xλyλ=−4+8λ,for1≤λ≤n,Ψ∗yλyλ+1=1+8λ,for1≤λ≤n−1,andΨ∗xλxλ+1=−1+8λ,for1≤λ≤n−1.
As a result,Ψ is an Cexp average assignment and the diamond ladder graph Dln is an Cexp average assignment graph, for n≥1.Theorem 10.
The latitude graph is anCexp average assignment graph.Proof.
Make the vertex assignment;Ψ:VG⟶N−3n/2+1,3n/2+2,…,∞,(24)Ψuλ=−2+3λ,1≤λ≤n2,−1+3λ1,λ=n2,3n2,λ=n2+1,3+3n−3λ,n2+2≤λ≤n−1,3,λ=n.
Consequently, the instigated edge assignmentΨ∗ is acquired as follows:(25)Ψ∗uλun+2−λ=−2+3λ,for2≤λ≤n2,Ψ∗unu1=2,andΨ∗uλuλ+1=3λ,1≤λ≤n2,−1+3n2,λ=n2+1,2+3n−3λ,n2+2≤λ≤n−1.
As a result,Ψ is an Cexp average assignment and the latitude graph is an Cexp average assignment graph.


## 5. Conclusion

The significant properties onCexp average assignment of several ladder networks are discovered in this work. Analysis of the Cexp average assignment of other networks can be discussed further.



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