Even Vertex Odd Mean Labeling of Transformed Trees

Abstract

Let G=(V,E) be a graph with p vertices and q edges. A graph G is said to have an even vertex odd mean labeling if there exists a function f:V(G)→{0,2,4,…,2q} satisfying f is 1-1 and the induced map f^*:E(G)→{1,3,5,…,2q-1} defined by f^* (uv)=(f(u)+f(v))/2 is a bijection. A graph that admits even vertex odd mean labeling is called an even vertex odd mean graph. In this paper, we prove that Tp-tree T , T@P_n , T@〖2P〗_n and 〈TõK_(1,n) 〉 ,where T is a Tp-tree, are even vertex odd mean graphs.

Full text

TWMS J. App. Eng. Math. V.10, N.2, 2020, pp. 338-345
1. Introduction
Throughout this paper by a graph we mean a finite, simple and undirected one. The
vertex set and the edge set of a graph Gare denoted by V(G) and E(G) respectively. A
graph labeling is an assignment of integers to the vertices or edges or both, subject to
certain conditions. Terms and notations not defined here are used in the sense of Harary
[3]. There are several types of labeling. An excellent survey of graph labeling is available
in [2]. The concept of mean labeling was introduced by Somasundaram and Ponraj [5].
A graph G(V, E ) with pvertices and qedges is called a mean graph if there is an
injective function fthat maps V(G) to {0,1,2, ..., q}such that for each edge uv, labeled
with f(u)+f(v)
2if f(u)+f(v) is even and f(u)+f(v)+1
2if f(u)+f(v) is odd. Then the resulting
edge labels are distinct. The notion odd mean labeling was introduced by Manickam and
Marudai in [4].
1Research Centre, Department of Mathematics, Govindammal Aditanar College for Women,
Tiruchendur-628 215, Tamilnadu, India.
e-mail: jeyajeyanthi@rediffmail.com; ORCID: https://orcid.org/0000-0003-4349-164X.
2Department of Mathematics, Government Arts College for Women, Ramanathapuram, Tamilnadu,
India.
e-mail: aymar padma@yahoo.co.in; ORCID: https://orcid.org/0000-0002-9261-4813.
3Research Scholar, Reg. No.: 12208, Manonmaniam Sundaranar University, Abishekapatti,
Tirunelveli-627 012, Tamilnadu, India.
e-mail: selvm80@yahoo.in; ORCID: https://orcid.org/0000-0002-6573-676X.
§Manuscript received: August 18, 2018; accepted: August 24, 2019.
TWMS Journal of Applied and Engineering Mathematics, Vol.10, No.2 c
I¸sık University, Department
of Mathematics, 2020; all rights reserved.
338
AMS
Subject
Classification:
05C78
even
vertex
odd
mean
graph.
Keywords:
mean
labeling,
odd
mean
labeling,
Tp−tree,
even
vertex
odd
mean
labeling,
odd
mean
graphs.
(transformed
tree),
T
@Pn,
T
@2Pn
and
hT
˜oK1,ni
(where
T
is
a
Tp-tree),
are
even
vertex
labeling
is
called
an
even
vertex
odd
mean
graph.
In
this
paper,
we
prove
that
Tp-tree
2
is
a
bijection.
A
graph
that
admits
even
vertex
odd
meandefined
by
f∗(uv)
=
f(u)+f(v)
{0,
2,
4,
...,
2q}
satisfying
f
is
1-1
and
the
induced
map
f∗
:
E(G)
→
{1,
3,
5,
...,
2q
−
1}
said
have
an
even
vertex
odd
mean
labeling
if
there
exists
a
function
f
:
V
(G)
→
Abstract.
Let
G
=
(V,
E)
be
a
graph
with
p
vertices
and
q
edges.
A
graph
G
is
EYANTHI1,
D.
RAMYA2,
M.
SELVI3, §P.J
EVEN
VERTEX
ODD
MEAN
LABELING
OF
TRANSFORMED
TREES

P. JEYANTHI, D. RAMYA, M. SELVI: EVEN VERTEX ODD MEAN LABELING OF ... 339
Let G(V, E ) be a graph with pvertices and qedges. A graph Gis said to be odd
mean if there exists a function f:V(G)→ {0,1,2,3, ..., 2q−1}satisfying fis 1-1 and the
induced map f∗:E(G)→ {1,3,5, ..., 2q−1}defined by
f∗(uv) = (f(u)+f(v)
2if f(u) + f(v) is even
f(u)+f(v)+1
2if f(u) + f(v) is odd
The concept of even vertex odd mean labeling was introduced in [6]. Let G(V , E)
be a graph with pvertices and qedges. A graph Gis said have an even vertex odd mean
labeling if there exists a function f:V(G)→ {0,2,4, ..., 2q}satisfying fis 1-1 and the
induced map f∗:E(G)→ {1,3,5, ..., 2q−1}defined by f∗(uv) = f(u)+f(v)
2is a bijection.
A graph that admits even vertex odd mean labeling is called an even vertex odd mean
graph. Motivated by the concepts of even vertex odd mean labeling [6] and Tp-tree [1], in
this paper we prove that Tp-tree, T@Pn,T@2PnhT˜oK1,niadmit even vertex odd mean
labeling.
We use the following definitions in the subsequent sequel.
2. Definition
Definition 2.1. Let Tbe a graph and uoand vobe two adjacent vertices in V(T). Let
there be two pendant vertices uand vin Tsuch that the length of uo-upath is equal
to the length vo-vpath. If the edge uovois deleted from Tand u,vare joined by an
edge uv, then such a transformation of Tis called an elemantary parallel transformation
(or an EPT) and the edge uovois called a transformable edge. If by a sequence of EPT’s
Tcan be reduced to a path, then Tis called a Tp-tree (transformed tree) and any such
sequence regarded as a composition of mappings (EPT’s) denoted by P, is called a parallel
transformation of T. The path, the image of Tunder Pis denoted as P(T).
Definition 2.2. Let Tbe a Tp-tree with mvertices. Let T@Pnbe the graph obtained from
Tand mcopies of Pnby identifying one pendant vertex of ith copy of Pnwith ith vertex
of T, where Pnis a path of length n−1. Let T@2Pnbe the graph obtained from Tby
identifying the pendant vertices of two vertex disjoint paths of equal lengths n−1at each
vertex of the Tp-tree T.
Definition 2.3. Let Tbe a Tp-tree with mvertices. Let hT˜oK1,nibe a graph obtained
from Tand mcopies of K1,n by joining the central vertex of ith copy of K1,n with ith
vertex of Tby an edge.
3. Even Vertex Odd Mean Labeling of Transformed Trees
Theorem 3.1. Every Tp-tree Tis an even vertex odd mean graph.
Proof. Let Tbe a Tp-tree with nvertices.
By the definition of Tp-tree there exists a parallel transformation Pof Tsuch that for the
path P(T) we have (i) V(P(T)) = V(T) and
(ii) E(P(T)) = (E(T)\Ed)∪EP,
where Edis the set of edges deleted from Tand EPis the set of edges newly added through
the sequence P= (P1, P2, ..., PK) of the EPT’s Pused to arrive at the path P(T). Clearly
Edand Ephave the same number of edges. Now, denote the vertices of P(T) successively
as v1, v2, v3, ..., vnstarting from one pendant vertex of P(T) right up to the other.
Define f:V(T)→ {0,2,4, ..., 2q}as follows:
f(vi) = 2(i−1) for 1 ≤i≤n.

340 TWMS J. APP. ENG. MATH. V.10, N.2, 2020
Let vivjbe a transformed edge in Tfor some indices iand j, 1 ≤i≤j≤mand P1be
the EPT that deletes the edge vivjand adds the edge vi+tvj−twhere tis the distance of
vifrom vi+tand also the distance of vjfrom vj−t.
Let Pbe a parallel transformation of Tthat contains P1as one of the constituent EPT’s.
Since vi+tvj−tis an edge in the path P(T), i+t+ 1 = j−twhich implies j=i+ 2t+ 1.
The induced label of the edge vivjis given by,
f∗(vivj) = f∗(vivi+2t+1) = f(vi)+f(vi+2t+1)
2= 2(i+t)−1 ......(1)
and f∗(vi+tvj−t) = f∗(vi+tvi+t+1) = f(vi+t)+f(vi+t+1)
2= 2(i+t)−1 .....(2)
Therefore from (1) and (2), f∗(vivj) = f∗(vi+tvj−t).
Let ej=vjvj+1 for 1 ≤j≤n−1.
For the vertex labeling f, the induced edge label f∗is defined as follows:
f∗(ej)=2j−1 for 1 ≤j≤n−1.
Therefore, fis an even vertex odd mean labeling of T.
Hence, Tis an even vertex odd mean graph.
For example, an even vertex odd mean labeling of a TP-tree with 14 vertices is given in
Figure 1. 
Figure 1. TP-tree with 14 vertices.
Theorem 3.2. Let Tbe a Tp-tree with mvertices. Then the graph T@Pnis an even
vertex odd mean graph.
Proof. Let Tbe a Tp-tree with mvertices. By the definition of a Tp-tree there exists a
parallel transformation Pof Tsuch that for the path P(T) we have (i) V(P(T)) = V(T)
and (ii) E(P(T)) = (E(T)\Ed)∪Ep, where Edis the set of edges deleted from Tand Ep
is the set of edges newly added through the sequence P= (P1, P2, ..., PK) of the EPT’s
Pused to arrive at the path P(T). Clearly Edand Ephave the same number of edges.
Now denote the vertices of P(T) successively as v1, v2, v3, ..., vmstarting from one pendant
vertex of P(T) right up to other. Let uj
1, uj
2, uj
3, ..., uj
n(1 ≤j≤m) be the vertices of jth
copy of Pn. Then V(T@Pn) = {uj
i: 1 ≤i≤n, 1≤j≤mwith uj
n=vj}.
Define f:V(T@Pn)→ {0,2,4, ..., 2q}as follows:
f(uj
i)=2n(j−1) + 2(i−1) if jis odd, 1 ≤i≤n, 1 ≤j≤m,
f(uj
i)=2n(j−1) + 2(n−i) if jis even, 1 ≤i≤n, 1 ≤j≤m.
Let vivjbe a transformed edge in Tfor some indices iand j, 1 ≤i≤j≤mand P1be

P. JEYANTHI, D. RAMYA, M. SELVI: EVEN VERTEX ODD MEAN LABELING OF ... 341
the EPT that deletes the edge vivjand adds the edge vi+tvj−twhere tis the distance of
vifrom vi+tand also the distance of vjfrom vj−t.
Let Pbe a parallel transformation of Tthat contains P1as one of the constituent EPT’s.
Since vi+tvj−tis an edge in the path P(T), i+t+ 1 = j−twhich implies j=i+ 2t+ 1.
The induced label of the edge vivjis given by,
f∗(vivj) = f∗(vivi+2t+1) = f(vi)+f(vi+2t+1)
2= 2n(i+t)−1 ......(3)
f∗(vi+tvj−t) = f∗(vi+tvi+t+1) = f(vi)+f(vi+2t+1)
2= 2n(i+t)−1......(4)
Therefore from (3) and (4), f∗(vivj) = f∗(vi+tvj−t).
Let ej
i=uj
iuj
i+1 for 1 ≤i≤n−1, 1 ≤j≤mand ej=vjvj+1 for 1 ≤j≤m−1.
For the vertex labeling f, the induced edge label f∗is defined as follows:
f∗(ej
i)=2n(j−1) + 2i−1 if jis odd, 1 ≤i≤n−1, 1 ≤j≤m,
f∗(ej
i) = 2(nj −i)−1 if jis even, 1 ≤i≤n−1, 1 ≤j≤m,
f∗(ej)=2nj −1 for 1 ≤j≤m−1.
Therefore,fis an even vertex odd mean labeling of T@Pn. Hence T@Pnis an even vertex
odd mean graph. For example, an even vertex odd mean labeling of T@P4, where Tis a
Tp-tree with 8 vertices, is given in Figure 2. 
Figure 2. T@P4
Theorem 3.3. Let Tbe a Tp-tree with mvertices. Then the graph T@2Pnis an even
vertex odd mean graph.
Proof. Let Tbe a Tp-tree with mvertices. By the definition of a Tp-tree there exists a
parallel transformation Pof Tsuch that for the path P(T) we have (i) V(P(T)) = V(T)
and (ii) E(P(T)) = (E(T)\Ed)∪Ep, where Edis the set of edges deleted from Tand Ep
is the set of edges newly added through the sequence P= (P1, P2, ..., PK) of the EPT’s P
used to arrive at the path P(T). Clearly Edand Ephave the same number of edges.
Now denote the vertices of P(T) successively as v1, v2, v3, ..., vmstarting from one pendant
vertex of P(T) right up to other. Let uj
1,1, uj
1,2, uj
1,3, ..., uj
1,n and uj
2,1, uj
2,2, uj
2,3, ..., uj
2,n

342 TWMS J. APP. ENG. MATH. V.10, N.2, 2020
(1 ≤j≤m) be the vertices of the two vertex disjoint paths identified with jth vertex of T
such that vj=uj
1,n =uj
2,n. Then V(T@2Pn) = {vj, uj
1,i, uj
2,i : 1 ≤i≤n, 1≤j≤mwith
vj=uj
1,n =uj
2,n}. Define f:V(T@2Pn)→ {0,2,4, ..., 2q}as follows:
f(uj
1,i) = 2((2n−1)j−2n+i) for 1 ≤i≤n, 1 ≤j≤m,
f(uj
2,i) = 2((2n−1)j−i) for 1 ≤i≤n, 1 ≤j≤m.
Let vivjbe a transformed edge in Tfor some indices iand j, 1 ≤i≤j≤mand P1be
the EPT that deletes the edge vivjand adds the edge vi+tvj−twhere tis the distance of
vifrom vi+tand also the distance of vjfrom vj−t.
Let Pbe a parallel transformation of Tthat contains P1as one of the constituent EPT’s.
Since vi+tvj−tis an edge in the path P(T), i+t+ 1 = j−twhich implies j=i+ 2t+ 1.
The induced label of the edge vivjis given by,
f∗(vivj) = f∗(vivi+2t+1) = f(vi)+f(vi+2t+1)
2= 2(2n−1)(i+t)−1 ......(5)
f∗(vi+tvj−t) = f∗(vi+tvi+t+1) = f(vi)+f(vi+2t+1)
2= 2(2n−1)(i+t)−1......(6)
Therefore from (5) and (6), f∗(vivj) = f∗(vi+tvj−t).
ej
1,i =uj
1,iuj
1,i+1 for 1 ≤i≤n−1, 1 ≤j≤m
ej
2,i =uj
2,iuj
2,i+1 for 1 ≤i≤n−1, 1 ≤j≤mand
ej=vjvj+1 for 1 ≤j≤m−1.
For the vertex labeling f, the induced edge label f∗is defined as follows:
f∗(vjvj+1)=2j(2n−1) −1 for 1 ≤j≤m−1,
f∗(ej
1,i) = 2(2n−1)j+ 2i−4n+ 1 for 1 ≤i≤n−1, 1 ≤j≤m,
f∗(ej
2,i) = 2(2n−1)j−2i−1 for 1 ≤i≤n−1, 1 ≤j≤m,
Therefore,fis an even vertex odd mean labeling of T@2Pn.
Hence T@2Pnis an even vertex odd mean graph.
For example, an even vertex odd mean labeling of T@2P3, where Tis a Tp-tree with 12
vertices, is given in Figure 3. 
Theorem 3.4. Let Tbe a Tp-tree with 2mvertices. Then the graph hT˜oK1,niis an even
vertex odd mean graph.
Proof. Let Tbe a Tp-tree with 2mvertices.
By the definition of Tp-tree there exists a parallel transformation Pof Tsuch that for the
path P(T), we have (i) V(P(T)) = V(T) and (ii) E(P(T)) = (E(T)\Ed)∪Ep,
where Edis the set of edges deleted from Tand Epis the set of edges newly added through
the sequence P= (P1, P2, ..., PK) of the EPT’s Pused to arrive at the path P(T).
Clearly Edand Ephave the same number of edges.
Now denote the vertices of P(T) successively as v1, v2, v3, ..., v2mstarting from one pendant
vertex of P(T) right up to other.
Let ui
0, ui
1, ui
2, ..., ui
nbe the vertices of the ith copy of K1,n, attached with viof Tby an
edge.
Define f:V(hT˜oK1,n i)→ {0,2,4, ..., 2q}as follows:
f(vj) = 2(n+ 2)(j−1) if jis odd and 1 ≤j≤2m,
f(vj) = 2(n+ 2)(j−2) + 4n+ 6 if jis even and 1 ≤j≤2m,
f(uj
0) = 2(n+ 2)(j−1) + 2 if jis odd and 1 ≤j≤2m,
f(uj
0) = 2(n+ 2)(j−2) + 4n+ 4 if jis even and 1 ≤j≤2m,
f(uj
i) = 2(n+ 2)(j−1) + 4iif jis odd and 1 ≤j≤2m, 1 ≤i≤n,
f(uj
i) = 2(n+ 2)(j−2) + 4i+ 2 if jis even and 1 ≤j≤2m, 1 ≤i≤n.
Let vivjbe a transformed edge in Tfor some indices iand j, 1 ≤i≤j≤2mand

P. JEYANTHI, D. RAMYA, M. SELVI: EVEN VERTEX ODD MEAN LABELING OF ... 343
Figure 3. T@2P3
P1be the EPT that deletes the edge vivjand adds the edge vi+tvj−t
where tis the distance of vifrom vi+tand also the distance of vjfrom vj−t.
Let Pbe a parallel transformation of Tthat contains P1as one of the constituent EPT’s.
Since vi+tvj−tis an edge in the path P(T), i+t+ 1 = j−twhich implies j=i+ 2t+ 1.
The induced label of the edge vivjis given by,
f∗(vivj) = f∗(vivi+2t+1) = f(vi)+f(vi+2t+1)
2= 2(n+ 2)(i+t−1) + 2n+ 3 ......(7)
f∗(vi+tvj−t) = f∗(vi+tvi+t+1) = f(vi)+f(vi+2t+1 )
2= 2(n+ 2)(i+t−1) + 2n+ 3 ......(8).
Therefore from (7) and (8), f∗(vivj) = f∗(vjvj−t).
Let ej
i=uj
0uj
ifor 1 ≤j≤2m, 1 ≤i≤n.
For the vertex labeling f, the induced edge label f∗is defined as follows:
f∗(vjvj+1) = 2(n+ 2)(j−1) + 2n+ 3 for 1 ≤j≤2m−1,
f∗(vjuj
0) = 2(n+ 2)(j−1) + 1 if jis odd and 1 ≤j≤2m,
f∗(vjuj
0) = 2(n+ 2)j−3 if jis even and 1 ≤j≤2m,
f∗(uj
0uj
i) = 2(n+ 2)(j−1) + 2i+ 1 if jis odd and 1 ≤j≤2m, 1 ≤i≤n,
f∗(uj
0uj
i) = 2(n+ 2)(j−2) + 2(n+i) + 3 if jis even and 1 ≤j≤2m,1 ≤i≤n.
Therefore, fis an even vertex odd mean labeling of hT˜oK1,n i.
Hence hT˜oK1,n iis an even vertex odd mean graph.
For example, an even vertex odd mean labeling of hT˜oK1,5i, where Tis a Tp-tree with 8
vertices, is given in Figure 4. 

344 TWMS J. APP. ENG. MATH. V.10, N.2, 2020
Figure 4. hT˜oK1,5i
References
[1] Acharya, B. D., (2004), Parallel Transformation of graphs: Graphs having Unique Elementary Parallel
Transformation, AKCE J. Graphs and Combin., (1) pp. 63-67.
[2] Gallian, J. A., (2017), A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics,
#DS6.
[3] Harary, F., (1972), Graph theory, Addison Wesley, Massachusetts.
[4] Manickam, K., Marudai, M., (2006), Odd mean labeling of graphs, Bulletin of Pure and Applied
Sciences, 25E(1) pp. 149-153.
[5] Somasundaram, S., Ponraj, R., (2003), Mean labelings of graphs, National Academy Science Letter,
(26) pp. 210-213.
[6] Vasuki, R., Nagarajan, A., Arockiaraj, S., (2013), Even vertex odd mean labeling of graphs, SUT
Journal of Mathematics, 49(2) pp.79-92.

P. JEYANTHI, D. RAMYA, M. SELVI: EVEN VERTEX ODD MEAN LABELING OF ... 345
Dr. P. Jeyanthi, is the Principal of Govindammal Aditanar College for Women,
Tiruchendur,Tamilnadu. India. Under her guidance, 12 scholars have been awarded
Ph.D. degree. She is a referee for 30 International and 10 Indian journals and reviewer
of ‘Mathematical Reviews’ USA. She served as ‘Managing Editor’ of ‘International
Journal of Mathematics and Soft Computing’ (2011 to 2018). She has published
research papers 30 in Indian and 122 in foreign journals. Citation Index : Citations
:664 h-index:12, i-10 Index-20. Research Gate Score is 22.81. She is the author of
“Studies in Graph Theory - Magic labeling and Related Concepts”
Dr. D. Ramya, is working as an Assistant Professor, Department of Mathematics,
Government Arts College for Women, Ramanathapuram, Tamilnadu, India. She
has total teaching experience of 19 years. She did her Ph.D under the guidance of
Dr.P.Jeyanthi and has been awarded Ph.D Degree in the year 2010. She has published
22 research papers and 2 more papers are accepted for publication in journals with the
impact factor. She has participated in 2 national level conferences and also presented
a paper in one International level conference. Further, she served as a referee for the
international journal ‘Utilitas Mathematica’. Her Research Gate Score is 7.12.
M. Selvi, is working as an Assistant Professor in the Department of Mathematics,
Padmashri Dr.Sivanthi Aditanar College of Engineering,Tiruchendur,Tamilnadu. She
has total teaching experience of 8 years and started her Ph.D research work in May
2016 under the guidance of Dr.P.Jeyanthi. She has published 6 research papers in
well refereed foreign journals and 2 more papers are accepted for publication in the
journals with the impact factor. She has participated in one National seminar and
one State level seminar.