Skolem Difference Mean Graphs

Abstract

A graph G = (V, E) with p vertices and q edges is said to have skolem difference mean labeling if it is possible to label the vertices x ∈ V with distinct elements f (x) from 1, 2, 3, · · · , p+q in such a way that for each edge e = uv, let f*(e)= ⌈|f(u)-f(v)|/2⌉ and the resulting labels of the edges are distinct and are from 1, 2, 3, · · ·, q. A graph that admits a skolem difference mean labeling is called a skolem difference mean graph. In this paper, we prove Cn@Pm(n ≥ 3, m ≥ 1), are skolem difference mean graphs.

Full text

Skolem Difference Mean Graphs
M. Selvi
D. Ramya
Dr. Sivanthi Aditanar College of Engineering, India
and
P. Jeyanthi
Govindammal Aditanar College for Women, India
Received : April 2015. Accepted : June 2015
Proyecciones Journal of Mathematics
Vol. 34, No3, pp. 243-254, September 2015.
Universidad Cat´olica del Norte
Antofagasta - Chile
Abstract
A graph G=(V, E)with pvertices and qedges is said to have
skolem difference mean labeling if it is possible to label the vertices
x∈Vwith distinct elements f(x)from 1,2,3,···,p+qinsuchaway
that for each edge e=uv,letf∗(e)=l|f(u)−f(v)|
2mand the resulting
labels of the edges are distinct and are from 1,2,3,···,q.Agraph
that admits a skolem difference mean labeling is ca lled a skolem dif-
ferencemeangraph. Inthispaper,weproveCn@Pm(n≥3,m ≥1),
ThK1,n1:K1,n2:···:K1,nmi,ThK1,n1◦K1,n2◦◦◦K1,nmi,
St(n1,n
2,···,n
m)and Bt(n, n, ···,n
|{z }
mtimes
)are skolem difference mean graphs.
Keywords : Mean labeling, skolem difference mean labeling, skolem
difference mean graph, extra skolem difference mean labeling.
AMS Subject Classification : 05C78

244 M. Selvi, D. Ramya and P. Jeyanthi
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. Terms and notations not defined here are used in
the sense of Harary[1]. A graph labeling is an assignment of integers to the
vertices or edges or both, subject to certain conditions. 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
[8]. A graph G=(V, E)withpvertices and qedges is called a mean graph if
there is an injective function fthat maps V(G)to{0,1,2,···,q}such that
each edge uv is labeled with f(u)+f(v)
2if f(u)+f(v)isevenandf(u)+f(v)+1
2
if f(u)+f(v) is odd. Then the resulting edge labels are distinct. The notion
of skolem difference mean labeling was due to Murugan and Subramanian
[3]. A graph G=(V, E)withpvertices and qedgesissaidtohaveskolem
difference mean labeling if it is possible to label the vertices x∈Vwith
distinct elements f(x)from1,2,3,···,p+qin such a way that for each edge
e=uv,letf∗(e)=l|f(u)−f(v)|
2mand the resulting labels of the edges are
distinct and are from 1,2,3,···,q. A graph that admits a skolem difference
mean labeling is called a skolem difference mean graph. Further studies on
skolem difference mean labeling are available in [4]-[7].
In this paper, we extend the study on skolem difference mean labeling
and prove that Cn@Pm(n≥3,m ≥1), ThK1,n1:K1,n2:K1,n3:···:K1,nmi,
ThK1,n1◦K1,n2◦◦◦K1,nmi,St(n1,n
2,···,n
m)andBt(n, n, ···,n
|{z }
mtimes
)are
skolem difference mean graphs.
We use the following definitions in the subsequent section.
Definition 1.1. The graph Cn@Pmis obtained by identifying one pendant
vertex of the path Pmwith a vertex of the cycle Cn.
Definition 1.2. The shrub St(n1,n
2,···,n
m)is a graph obtained by con-
necting a vertex v0to the central vertex of each of mnumber of stars.
Definition 1.3. The banana tree Bt(n1,n
2,···,n
m
|{z }
mtimes
)is a graph obtained
by connecting a vertex v0tooneleafofeachofmnumber of stars.
Definition 1.4. Let G=(V, E)be a skolem difference mean graph with
pvertices and qedges. If one of the skolem difference mean labeling of G

Skolem Difference Mean Graphs 245
satisfies the condition that all the labels of the vertices are odd, then we
call this skolem difference mean labeling an extra skolem difference mean
labeling and call the graph Gan extra skolem difference mean graph [7].
An extra skolem difference mean labeling of P6isgiveninFigure1
2. Main Results
Theorem 2.1. The graph Cn@Pm(n≥3,m ≥1) is a skolem difference
mean graph.
Proof. Case (i): nis odd.
Let n=2k+1. Let u1,u
2,···,u
k,v
k,v
k−1,···,v
1,v
0be the vertices of
C2k+1 and let w1,w
2,···,w
mbe the vertices of Pm.ThenCn@Pmis ob-
tained by identifying w1of Pmwith v0of C2k+1, which has n+m−1
edges. E(C2k+1@Pm)={vivi+1|1≤i≤k−1}∪{uiui+1|1≤i≤k−1}∪
{wjwj+1|1≤j≤m−1}∪{v0v1,v
0u1,u
kvk}.Define f:V(C2k+1@Pm)→
{1,2,3,···,p+q=2n+2m−2}as follows:
f(w2i−1)=2i−1for1≤i≤§m
2¨,
f(w2i)=2m+3−2ifor 1 ≤i≤¥m
2¦,
f(u2i−1)=2m+2n+2−4ifor 1 ≤i≤lk
2m,
f(u2i)=4ifor 1 ≤i≤jk
2k,
f(v2i−1)=2m+2n+1−4ifor 1 ≤i≤lk
2m,
f(v2i)=4i+2for 1≤i≤jk
2k.
For each vertex label f, the induced edge label f∗is calculated as follows:
f∗(wiwi+1)=m+1−ifor 1 ≤i≤m−1,
f∗(ukvk)=1,
f∗(uiui+1)=n+m−1−2ifor 1 ≤i≤k−1,

246 M. Selvi, D. Ramya and P. Jeyanthi
f∗(vivi+1)=n+m−2−2ifor 1 ≤i≤k−1,
f∗(u1v0)=n+m−1, f∗(v1v0)=n+m−2.
Case (ii): nis even.
Let n=2k,k>1. Let v0,v
1,v
2,···,v
k−1,u
0,u
k−1,u
k−2,···,u
2,u
1be the
vertices of C2kand let w1,w
2,···,w
mbe the vertices of Pm.ThenCn@Pm
is obtained by identifying w1of Pmwith v0of C2k, which has n+m−1
edges. E(C2k@Pm)={vivi+1|1≤i≤k−2}∪{uiui+1|1≤i≤k−2}∪
{wjwj+1|1≤j≤m−1}∪{v0v1,v
0u1,u
0uk−1,u
0vk−1}.
Subcase (i): k>1 is odd.
Define f:V(C2k@Pm)→{1,2,3,···,p+q=2n+2m−2}as follows:
f(w2i−1)=2i−1for1≤i≤§m
2¨,
f(w2i)=2m+1−2ifor 1 ≤i≤¥m
2¦,
f(u2i−1)=2m+2n+2−4ifor 1 ≤i≤k−1
2,
f(u2i)=4ifor 1 ≤i≤k−1
2,
f(v2i−1)=2m+2n+1−4ifor 1 ≤i≤k−1
2,
f(v2i)=4i+2for 1≤i≤k−1
2,andf(u0)=2(n+m−k).
For each vertex label f, the induced edge label f∗is calculated as follows:
f∗(wiwi+1)=m−ifor 1 ≤i≤m−1,
f∗(uiui+1)=n+m−1−2ifor 1 ≤i≤k−2,
f∗(vivi+1)=n+m−2−2ifor 1 ≤i≤k−2,
f∗(v0u1)=n+m−1,
f∗(v0v1)=n+m−2,
f∗(uk−1u0)=m+1,
f∗(vk−1u0)=m.
Subcase (ii): k>1iseven.
Define f:V(C2k@Pm)→{1,2,3,···,p+q=2n+2m−2}as follows:
f(w2i−1)=2i−1for1≤i≤§m
2¨,
f(w2i)=2m+1−2ifor 1 ≤i≤¥m
2¦,
f(u2i−1)=2m+2n+2−4ifor 1 ≤i≤k
2,
f(u2i)=4ifor 1 ≤i≤k−2
2,
f(v2i−1)=2m+2n−4ifor 1 ≤i≤k
2,
f(v2i)=4i+1for 1≤i≤k−2
2,
and f(u0)=2k.
For each vertex label f, the induced edge label f∗is calculated as follows:
f∗(wiwi+1)=m−ifor 1 ≤i≤m−1,
f∗(uiui+1)=n+m−1−2ifor 1 ≤i≤k−2,
f∗(vivi+1)=n+m−2−2ifor 1 ≤i≤k−2,

Skolem Difference Mean Graphs 247
f∗(v0u1)=n+m−1,
f∗(v0v1)=n+m−2,
f∗(uk−1u0)=m+1,
f∗(vk−1u0)=m.
It can be verified that fis a skolem difference mean labeling. Hence Cn@Pm
is a skolem difference mean graph. 2
Skolem difference mean labelings of C7@P5and C10@P5are shown in
Figure 2.
In the following theorem, we prove that the graph
ThK1,n1:K1,n2:K1,n3:···:K1,nmi, obtained from the stars
K1,n1,K
1,n2,...,K
1,nmby joining the central vertices of K1,njand K1,nj+1
toanewvertexwjfor 1 ≤j≤m−1isanextraskolemdifference mean
graph.
Theorem 2.2. The graph ThK1,n1:K1,n2:K1,n3:··· :K1,nmiis an extra
skolem difference mean graph.
Proof. Let uj
i(1 ≤i≤nj) be the pendant vertices and vj(1 ≤j≤m)
be the central vertex of the star K1,nj(1 ≤j≤m). Then

248 M. Selvi, D. Ramya and P. Jeyanthi
ThK1,n1:K1,n2:K1,n3:···:K1,nmiis a graph obtained by joining vjand
vj+1 to a new vertex wj(1 ≤j≤m−1) by an edge. Define
f:V(ThK1,n1:K1,n2:K1,n3:···:K1,nmi)→
½1,2,3,···,p+q=2 m
P
k=1
nk+4m−3¾as follows:
f(vj)=2 m
P
k=1
nk+4m−3−2(j−1) for 1 ≤j≤m,
f(w1)=2n1+1,
f(wj)=2n1+1+2
j
P
k=2
(nk+1) for 2 ≤j≤m−1,
f(u1
i)=2i−1for1≤i≤n1
f(uj
i)=2
j−1
P
k=1
(nk+1)+2i−1for1≤i≤njand 2 ≤j≤m.
For each vertex label f, the induced edge label f∗is calculated as follows:
Let ej
i(1 ≤i≤njand 1 ≤j≤m) be the edges joining the vertices vj
with uj
i.f∗(e1
i)= m
P
k=1
nk+2m−i−1for1≤i≤n1,
f∗(ej
i)=nj+nj+1 +···+nm+2m−2j−i+1 for 1 ≤i≤njand 2 ≤j≤m,
f∗(v1w1)=n2+n3+···+nm+2(m−1),
f∗(vjwj)=nj+1 +nj+2 +···+nm+2(m−j)for2≤j≤m−1,
f∗(w1v2)=n2+n3+···+nm+2m−3,
f∗(wjvj+1)=nj+1 +nj+2 +···+nm+2(m−j)−1for2≤j≤m−1.
It can be verified that ThK1,n1:K1,n2:K1,n3:···:K1,nmiis an extra skolem
difference mean labeling. Hence ThK1,n1:K1,n2:K1,n3:···:K1,nmiis an
extra skolem difference mean graph. 2
Corollary 2.3. The graph T*K1,n :K1,n :K1,n :···:K1,n
|{z }
mtimes +is an extra
skolem difference mean graph.
An extra skolem difference mean labeling of ThK1,7:K1,7:K1,7iis
showninFigure3.

Skolem Difference Mean Graphs 249
The graph ThK1,n1◦K1,n2◦◦◦K1,nmiis obtained from the stars
K1,n1,K
1,n2,···,K
1,nmbyjoiningaleafofK1,njand a leaf of K1,nj+1 to a
new vertex wj(1 ≤j≤m−1) by an edge.
Theorem 2.4. The graph ThK1,n1◦K1,n2◦K1,n3◦◦◦K1,nmiis an extra
skolem difference mean graph.
Proof. Let uj
i(1 ≤i≤nj) be the pendant vertices and vj(1 ≤j≤m)
be the central vertex of the star K1,nj(1 ≤j≤m). Then
ThK1,n1◦K1,n2◦K1,n3◦◦◦K1,nmiis a graph obtained by joining uj
njand
uj+1
1to a new vertex wj(1 ≤j≤m−1) by an edge. Define
f:V(ThK1,n1◦K1,n2◦K1,n3◦◦◦K1,nmi)→
½1,2,3,···,p+q=2 m
P
k=1
nk+4m−3¾as follows:
f(vj)=2 m
P
k=1
nk+4m−4j+1 for 1 ≤j≤m,
f(wj)=2 m
P
k=1
nk+4m−4j−1for1≤j≤m−1,
f(u1
i)=2i−1for1≤i≤n1
f(uj
i)=2
j−1
P
k=1
nk+2i−1for1≤i≤njand 2 ≤j≤m.
Let ej
i=vjuj
ifor 1 ≤i≤njand 1 ≤j≤m. For each vertex label f,the
induced edge label f∗is calculated as follows: f∗(e1
i)= m
P
k=1
nk+2m−i−1
for 1 ≤i≤n1,
f∗(ej
i)=nj+nj+1 +···+nm+2m−2j−i+1 for 2 ≤j≤m−1and

250 M. Selvi, D. Ramya and P. Jeyanthi
1≤i≤nj,
f∗(u1
n1w1)= m
P
k=1
nk+2m−2−n1,
f∗(uj
njwj)=nj+1 +nj+2 +···+nm+2(m−j)for2≤j≤m−1,
f∗(w1u2
1)=n2+n3+···+nm+2m−3,
f∗(wjuj+1
1)=nj+1 +nj+2 +···+nm+2m−2j−1for2≤j≤m−1.
It can be verified that ThK1,n1◦K1,n2◦K1,n3◦◦◦K1,nmiis an extra
skolem difference mean labeling. Hence ThK1,n1◦K1,n2◦K1,n3◦◦◦K1,nmi
is an extra skolem difference mean graph. 2
Corollary 2.5. The graph T*K1,n ◦K1,n ◦K1,n ◦◦◦K1,n
|{z }
mtimes +is a skolem
difference mean graph.
Askolemdifference mean labeling of ThK1,6◦K1,6◦K1,6◦K1,6iis shown
in Figure 4.

Skolem Difference Mean Graphs 251
Theorem 2.6. If Gis a graph having pvertices and qedges with q>p
then Gis not a skolem difference mean graph.
Proof. Let Gbe a (p, q)graphwithq>p. The minimum possible
vertex label of Gis 1 and the maximum possible vertex label of Gis p+q.
Therefore, the maximum possible edge label of Gis lp+q−1
2m<l2q−1
2m=q.
That is, Ghas no edge having the label qand hence Gis not a skolem
difference mean graph. 2
Corollary 2.7. ThecompletegraphKnis a skolem difference mean graph
if and only if n≤3.
Proof. K2is P2and K3is C3. The graph Knhas nvertices and n(n−1)
2
edges. For n≥4, n(n−1)
2>n. By theorem 2.6, Kn,n≥4isnotaskolem
difference mean graph. 2
Theorem 2.8. The shrub St(n1,n
2,···,n
m)is a skolem difference mean
graph.
Proof. Let v0,v
j,u
j
i(1 ≤j≤m, 1≤i≤nj) be the vertices of
St(n1,n
2,···,n
m). Then
E(St(n1,n
2,···,n
m)) = {v0vj|1≤j≤m}∪nvjuj
i|1≤i≤njo.
Define f:V(St(n1,n
2,···,n
m)) →(1,2,3,···,p+q=2 m
P
j=1
nj+2m+1
)
as follows:
f(v0)=2m+2 m
P
j=1
nj+1,
f(vj)=2j−1,1≤j≤m,
f(uj
i)=2(nj+nj+1 +···+nm)+2(j−i), 1 ≤i≤nj−1and1≤j≤m,
f(uj
nj)=2(nj+1 +nj+2 +···+nm)+2j+1 for 1≤j≤m−1,
f(um
nm)=2m+1.
Let ej
i=vjuj
ifor 1 ≤i≤njand 1 ≤j≤m. For each vertex label f,
the induced edge label f∗is defined as follows: f∗(v0vj)=m+n1+n2+
···+nm−j+1, 1 ≤j≤m
f∗(ej
i)=nj+nj+1 +···+nm−i+1, 1≤i≤njand 1 ≤j≤m.
It can be verified that St(n1,n
2,···,n
m)isaskolemdifference mean
graph. 2

252 M. Selvi, D. Ramya and P. Jeyanthi
The skolem difference mean labeling of St(2,5,5,3)isshowninFigure
5.
Theorem 2.9. The banana tree Bt(n, n, ···,n
|{z }
mtimes
)is a skolem difference mean
graph.
Proof. Let v0,v
j,u
j
i(1 ≤j≤m, 1≤i≤n) be the vertices of
Bt(n, n, ···,n
|{z }
mtimes
). Then E(Bt(n, n, ···,n
|{z }
mtimes
)) = nv0uj
1,v
juj
i|1≤i≤nand 1 ≤j≤mo.
Define f:V(Bt(n, n, ···,n
|{z }
mtimes
)) →{1,2,3,···,p+q=2m(n+1)+1}as fol-
lows:
f(v0)=2m(n+1)+1,
f(vj)=2m(n−1) + 4j−1, for 1 ≤j≤m,
f(uj
1)=2j−1, 1 ≤j≤m,
f(uj
i)=(2i−2)m+2j−1, for 2 ≤i≤n−1and1≤j≤m,
f(uj
n)=2m(n+1)−2(m−j)for1≤j≤m.Letej
i=vjuj
ifor 1 ≤i≤n
and 1 ≤j≤m. For each vertex label f, the induced edge label f∗is
calculated as follows: f∗(v0uj
1)=m(n+1)−j+1, for 1≤j≤m
f∗(ej
i)=m(n−i)+j,for1≤i≤n−1and1≤j≤m,
f∗(vjuj
n)=m−j+1for 1≤j≤m.

Skolem Difference Mean Graphs 253
It can be verified that Bt(n, n, ···,n
|{z }
mtimes
)isaskolemdifference mean graph.
2
Askolemdifference mean labeling of Bt(3,3,3,3,3,3,3,3) is shown in Fig-
ure 6.
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M. Selvi
Department of Mathematics
Dr. Sivanthi Aditanar College of Engineering,
Tiruchendur-628 215, Tamilnadu,
India
e-mail: selvm80@yahoo.in
D. Ramya
Department of Mathematics
Dr. Sivanthi Aditanar College of Engineering,
Tiruchendur-628 215, Tamilnadu,
India
e-mail : aymar padma@yahoo.co.in
and
P. Jeya nthi
Research Centre, Department of Mathematics
Govindammal Aditanar College for Women
Tiruchendur-628 215, Tamilnadu,
India
e-mail : jeyajeyanthi@rediffmail.com