Linear Cyclic Snakes as Super Vertex Mean Graphs

2018·0.19 MB·

by  A. Lourdusamy

International J.Math. Combin. Vol.1(2018), 109-126
Linear Cyclic Snakes as Super Vertex Mean Graphs
A.Lourdusamy and Sherry George
DepartmentofMathematics,ManonmaniamSundaranarUniversity
St. Xavier’sCollege(Autonomous),Palayamkottai,Tirunelveli-627002,India
[email protected]
Abstract: A super vertex mean labeling f of a (p,q) - graph G(V,E) is defined as an
injectionfromE totheset 1,2,3, ,p+q thatinducesforeachvertexvthelabeldefined
{ ··· }
by the rule fv(v) = Round Pe∈dE(vv)f(e) , where Ev denotes the set of edges in G that are
incident at the vertex v, suc(cid:16)h that the s(cid:17)et of all edge label and the induced vertex labels is
1,2,3, ,p+q . Allthe cycles, Cn,n 3 and n=4 are super vertex mean graphs. Our
{ ··· } ≥ 6
attempt in this paper is to show that all the linear cyclic snakes, including kC , are also
4
super vertex mean graphs, even though C is not an SVM graph. We also define the term
4
SuperVertex Mean numberof graphs.
Key Words: Super vertex mean label, Smarandachely super H-vertex mean labeling,
linear cyclic snakes, SVMnumber.
AMS(2010): 05C78.
§1. Introduction
The graphsconsideredhere will be finite, undirected and simple. The symbols V(G) and E(G)
will denote the vertex set and edge set of a graph G respectively. p and q denote the number
of vertices and edges of G respectively. A graph of order p and size q is often called a (p,q) -
graph.
A labeling of a graph G is an assignment of labels either to the vertices or edges. There
are varieties of vertex as well as edge labeling that are already in the literature. Mean labeling
was introduced by Somasundaram and Ponraj [14]. Super mean labeling was introduced by
D.Ramya et al.[10]. Some results on mean labeling and super mean labeling are given in [5, 6,
9, 10, 11, 13, 14] and [15]. Lourdusamy and Seenivasan [5] introduced vertex mean labeling as
an edge analogue of mean labeling.
Continuing on the same line and inspired by the above mentioned concepts, Lourdusamy
et al. [7] brought in a new extension of mean labeling, called Super vertex mean labeling of
graphs.
§2. Super Vertex Mean Labeling
Definition 2.1 A super vertex mean labeling f of a (p,q) - graph G(V,E) is defined as an
1ReceivedFebruary4,2017, AcceptedMarch3,2018.
110 A.LourdusamyandSherryGeorge
injection from E to the set 1,2,3, ,p+q that induces for each vertex v the label defined
{ ··· }
by the rule fv(v) = Round e∈Evf(e) , where E denotes the set of edges in G that are
d(v) v
P
incident at the vertex v, suc(cid:16)h that the s(cid:17)et of all edge labels and the induced vertex labels is
1,2,3, ,p+q .
{ ··· }
Furthermore, a super vertex mean labeling f on G is Smarandachely super H-vertex mean
labeling if the induced vertex labels on vertex in H is 1,2,3, ,p(H ) + q(H ) for each
′ ′ ′
{ ··· }
subgraph H =H in G, where p(H ) and q(H ) are respectively the order and the size of H .
′ ∼ ′ ′ ′
A graph that accepts super vertex mean labeling is called a super vertex mean (hereafter,
SVM) graph. The following results have already been proved in [7] and [8]. We use them for
our further study on the super vertex mean behavior of linear cyclic snakes. Before entering
into the results, we define the term cyclic snakes.
Definition 2.2 A kC -snake has been defined as a connected graph in which all the blocks are
n
isomorphic to the cycle C and the block-cut point graph is a path P, where P is the path of
n
minimum length that contains all the cut vertices of a kC -snake. Barrientos [13] has proved
n
that any kC -snake is represented by a string s ,s ,s , ,s of integers of length k 2,
n 1 2 3 k 2
··· − −
where the ith integer, s on the string is the distance between ith and i+1th cut vertices along
i
the path, P, from one extreme and is taken from S = 1,2,3, , n . A kC -snake is said
n { ··· 2 } n
to be linear if each integer s of its string is equal to n .
i 2 (cid:4) (cid:5)
(cid:4) (cid:5)
Remark 2.3 The strings obtained from both the extremes are assumed to be the same. A
linear cyclic snake, kC is obtained from k copies of C by identifying the vertex v in the
n n i,r+1
ith copy of C at a vertex v in the (i+1)th copy of C , where 1 i k 1 and n = 2r
n i+1,1 n
≤ ≤ −
or n=2r+1, depending upon whether n is even or odd respectively.
Note2.4 thatv =v for1 i k 1,andweconsiderthisvertexasv throughout
i,r+1 i+1,1 i+1,1
≤ ≤ −
this paper.
§3. Known Results
We have known results listed in the following.
(1) All the cycles except C are SVM graphs([8]);
4
(2) All odd cycles, C can be SVM labeled as many as n ways and every even cycle, C ,
n 2 n
except C can have n 1 types of SVM labelings([8]);
4 2 − (cid:4) (cid:5)
(3) A linear triangular snake, kC with k blocks is an SVM graph([7]).
(cid:4) (cid:5) 3
§4. Linear Cyclic Snakes of Higher Orders
We proceed to prove that other linear cyclic snakes too are super vertex mean graphs.
Theorem 4.1 Linear quadrilateral snakes, kC with k 2 blocks are SVM graphs.
4
≥
LinearCyclicSnakesasSuperVertexMeanGraphs 111
Proof LetkC bealinearquadrilateralsnakewithpverticesandq edges. Thenp=3k+1
4
andq =4k. Define f :E(kC ) 1,2,3, ,7k+1 to be f(u u )=7i if 1 i k 1and
4 i i+1
→{ ··· } ≤ ≤ −
7k+1 i=k. When 1 i k 1 and k 2,
≤ ≤ − ≥
1, if i=1, and j =1
7i 5, if 2 i k 1, and j =1
f(ei,j)=77ii,−−1, iiff 11≤≤ii≤≤kk−−11,, aanndd jj ==23
≤ ≤ −
7i−4, if 1≤i≤k−1, and j =4.
When i=k, k 2 and k is even,
≥
7k 6, if j =1,
−
7k 3, if j =2,
f(ei,j)=7k−−1, if j =3,
7k+1, if j =4.

When i=k, k 3 and k is odd,
≥
7k 5, if j =1,
−
7k 2, if j =2,
f(ei,j)=7k−+1, if j =3,
7k 4, if j =4.
 −
It can be easily verified that f is injective. The induced vertex labels are as follows:
When 1 i k 1 and k 2,
≤ ≤ − ≥
2, if i=1, and j =1,
7i 6, if 2 i k 1, and j =1,
fv(vi,j)=7i−−3, if 1≤≤i≤≤k−−1, and j =2,
7i 2, if 1 i k 1, and j =4.
 − ≤ ≤ −
When i=k, k 2, and k is even,
≥
7k 5, if j =1,
−
7k 4, if j =2,
fv(vi,j)=7k−−2, if j =3,
7k, if j =4.

112 A.LourdusamyandSherryGeorge
When i=k, k 3, and k is odd,
≥
7k 6, if j =1,
−
7k 3, if j =2,
fv(vi,j)=7k,− if j =3,
7k 1, if j =4.
 −
Itcanbeeasilyverifiedthatthesetofedgelabelsandinducedvertexlabelsis 1,2,3,...,7k+
{
1 in two following cases:
}
Case 1. When k is even,
f(E) = 1,6,7,3,9,13,14,10,16,20,21,27, ,7k 12,
{ ··· −
7k 8,7k 7,7k 11,7k 6,7k 3,7k 1,7k+1
− − − − − − }
and,
fv(V)= 2,4,8,5,11,15,12,18,22,19, ,7k 10,7k 5,7k 9,7k 4,7k 2,7k .
{ ··· − − − − − }
Therefore,
f(E) fv(V) = 1,2,3,4, ,7k 12,7k 11,7k 10,7k 9,7k 8,
∪ { ··· − − − − −
7k 7,7k 6,7k 5,17k 4,7k 3,7k 2,7k 2,7k,7k+1 .
− − − − − − − }
Case 2. When k is odd,
f(E) = 1,6,7,3,9,13, ,7k 12,7k 8,
{ ··· − −
7k 7,7k 11,7k 5,7k 2,7k+1,7k 4
− − − − − }
and,
fv(V)= 2,4,8,5, ,7k 10,7k 6,7k 9,7k 3,7k,7k 1 .
{ ··· − − − − − }
Therefore,
f(E) fv(V) = 1,2,3,4, ,7k 12,7k 11,7k 10,7k 9,7k 8,
∪ { ··· − − − − −
7k 7,7k 6,7k 5,17k 4,7k 3,7k 2,7k 2,7k,7k+1 .
− − − − − − − }
Thus it has been proved that the labeling f : E(kC ) 1,2,3, ,7k+1 is a super
4
→ { ··· }
vertex mean labeling.
2
Example 4.2 The super vertex-mean labelings of two linear quadrilateral snakes with 4 and
3 blocks are shown in figures 1 and 2 respectively.
LinearCyclicSnakesasSuperVertexMeanGraphs 113
5 12 19 28
3 7 10 14 17 21 29 27
2 8 15 23 26
1 6 9 13 16 20 22 25
4 11 18 24
Figure 1 A Super vertex-mean labeling of a linear quadrilateral snake with 4 blocks
5 12 20
3 7 10 14 17 22
2 8 15 21
1 6 9 13 16 19
4 11 18
Figure 2 A super vertex-mean labeling of a linear quadrilateralsnake with 3 blocks
Theorem 4.3 All linear pentagonal snakes, kC with k,k 2 blocks are SVM graphs.
5
≥
Proof Let kC be a linear pentagonal snake with k,k 2 blocks of C . Here p=4k+1,
5 5
≥
q =5k and p+q =9k+1. Define f :E(kC ) 1,2,3, ,9k+1 as follows:
5
→{ ··· }
When i=1,
5, if j =1,
2j+4, if 2 j 3,
f(ei,j)=1, if j =≤4,≤
3, if j =5.
And When 2 i k, 
≤ ≤
9i 9, if j =1,
−
f(ei,j)=9i+2j 5, if 2 j 3,
9i+3j−18, if 4≤j ≤5.
− ≤ ≤
It can be easily verified that f isinjective. Then, the induced vertex labels are as follows:
When i=1,
4, if j =1,
7, if j =2,
fv(vi,j)=6, if j =4,
2, if j =5.

114 A.LourdusamyandSherryGeorge
When 2 i k,
≤ ≤
9i+2j 9, if 1 j 2
− ≤ ≤
fv(vi,j)=9i 2j+6, if 4 j 5
9k−, if i=≤k,≤and j =3.

Clearly it can be proved that the union of the set of edge labels and the induced vertex
labelsis 1,2,3, ,9k+1 . Therefore,alllinearpentagonalsnakeskC aresupervertexmean
5
{ ··· }
graphs.
2
Example 4.4 A graphgivenin Figure 3 is anSVM labeling ofa linear pentagonalsnake with
3 blocks.
1
2 6
3
10 12
4 14 16
15
5 21
7 11 19 24 23 25
8
9
13 20 28
17
18
22 27
26
Figure 3 An SVM labeling of a linear pentagonal snake, 3C .
5
Theorem 4.5 All linear hexagonal snakes, kC ,k 2 are super vertex mean graphs.
6
≥
Proof Let kC be a hexagonal snake with k,k 2 blocks of C . p = 5k+1 and q = 6k
6 6
≥
and p+q =11k+1. Define f :E(G ) 1,2,3, ,11k+1 as follows:
n
→{ ··· }
3j, if i=1, and 1 j 4,
≤ ≤
7, if i=1, and j =5,
f(ei,j)=11,1i−4j, iiff i2=≤1i,≤ankd, ajn=d61,≤j ≤2,
11i+3j 11, if 2 i k, and 3 j 4,
11i−8j−+37, if 2≤≤i≤≤k, and 5≤≤j ≤≤6.
LinearCyclicSnakesasSuperVertexMeanGraphs 115
It can be easily verified that f is injective. Then, the induced vertex labels are as follows:
11i+3j 12, if 1 i k, and 1 j 2,
− ≤ ≤ ≤ ≤
8, if i=1, and j =3,
fv(vi,j)=1111ik−, 5, iiff i2=≤ki,≤akn,dajn=d4j,=3,
11i 1, if 1 i k, and j =5,
Clearly it can be provedth1a1ti−−th7e,union ofifth1e≤≤seit≤≤ofk,edagnedlajb=els6.and the induced vertex
labels is 1,2,3, ,11k+1 .
{ ··· }
Therefore, linear hexagonal snakes, all kC with k blocks of C are super vertex mean
6 6
graphs.
2
Example 4.6 A graph given in Figure 4 is an SVM labeling of a linear hexagonal snake with
3 blocks.
7 19 30
1 4 10 12 11 15 21 23 22 26 32 34
2 13 24 33
3 5 8 9 18 16 17 20 29 27 28 31
6 14 25
Figure 4 An SVM labeling of a linear hexagonal snake, 3C .
6
Theorem 4.7 All linear heptagonal snakes, kC ,k 2 are super vertex mean graphs.
7
≥
Proof Let kC be a linear heptagonal snake with k,k 2 blocks of C . p = 6k+1 and
7 7
≥
q =7k and p+q =13k+1. Define f :E(G ) 1,2,3, ,13k+1 as follows:
n
→{ ··· }
4j 1, if i=1, and 1 j 3,
− ≤ ≤
30 4j, if i=1, and 4 j 6,
f(ei,j)=11,3i−+2j−7, iiff i2=≤1i,≤ankd, ajn=≤d71,≤≤j ≤4,
13i 13, if 2 i k, and j =5,
13i−+2j−21, if 2≤≤i≤≤k, and 6≤j ≤7.
116 A.LourdusamyandSherryGeorge
It can be easily verified that f is injective. Then, the induced vertex labels are as follows:
2, if i=1, and j =1,
4j 3, if i=1, and 2 j 3,
fv(vi,j)=311233ii−−−+412jj0,,−8, iiifff i22=≤≤1ii,≤≤ankkd,, aa5nn≤≤ddjj2≤≤=≤71j,,≤3,
13i 6, if 2 i k, and j =5,
Clearly it can be provedth11a33tik−+,th3eju−ni2o9n,ofiiffthi2e=≤≤skeit,≤≤oafkn,dedajgne=dla46b.≤elsja≤nd7,the induced vertex
labels is 1,2,3, ,13k+1 .
{ ··· }
Therefore, all linear heptagonal snakes, kC with k blocks of C are super vertex mean
7 7
graphs.
2
Example 4.8 AgraphgiveninFigure5is anSVMlabeling ofa linearheptagonalsnake,3C .
7
6 10
1 4 8 12
17 13
2 1419 18 15 20
3 5 9 16 27
7 11
21 22 24 29
23 25 32
34
35 31
36
30
37 28
38 26
39 33
40
Figure 4 A Super Vertex Mean Labeling of 3C linear snake.
7
Theorem 4.9 Let kC be a linear cyclic snake with k,k 2 blocks of C ,n 8 and n
n n
≥ ≥ ≡
0(mod 2). Then kC is a super vertex mean graph.
n
Proof LetkC bealinearcyclicsnakewithk,k 2blocksofC ,n 8andn 0(mod2).
n n
≥ ≥ ≡
Let n = 2r,r 4. Now, p = (n 1)k +1 and q = nk and p+q = (2n 1)k +1. Define
≥ − −
LinearCyclicSnakesasSuperVertexMeanGraphs 117
f :E(kC ) 1,2,3, ,(2n 1)k+1 as follows:
n
→{ ··· − }
3j, if i=1, and 1 j 3,
≤ ≤
4j 3, if i=1, and 4 j r,
f(ei,j)=417((22n,,nn−−−411j))+ii−4,22nn++84,, iiiiifffff iii22===≤111ii,,,≤aaannnkkddd,, aarjjnn+==≤dd1nnjj,≤≤−==j112,,,≤n−2,
− − ≤ ≤
((((2222nnnn−−−−1111))))iii(i−−+−2221nnn),++−414jj0,−+25,, iiiiffff i222=≤≤≤kiii,≤≤≤akkkn,,,daaajnnn=dddnjr4.+=≤13j,≤≤jr,≤n−1,
And, the induced vertex labels are as follows:
2, if i=1, and j =1,
3j 1, if i=1, and 2 j 4,
fv(vi,j)=44((422j,nnn−−−−−54,11j))+ii−−6,22nn++36,, iiiiifffff iii22===≤≤111ii,,,≤≤aaannnkkddd,, aajr5nn+=≤≤dd2njjj,≤≤≤==jr12,,,≤n−1,
(2n 1)i 2n+7, if 2 i k, and j =3,
((((2222nnnn−−−−−1111))))iiki,−−−+222nnn++−445j,j−+47,, iiiiffff i222=≤≤≤≤kiii,≤≤≤≤akkkn,,,daaajnnn=dddrrj4++=≤12nj.,≤≤jr,≤n−1,
We prove the theorem by mathematical induction on r, where n = 2r,r 4. The above
≥
edge labeling function f(e) and the induced vertex labeling function fv(v) are expressed in
terms of r as follows:
3j, if i=1, and 1 j 3,
≤ ≤
4j 3, if i=1, and 4 j r,
f(ei,j)=87r, −−4j+4, iiff ii==11,, aanndd rj+=≤12r≤≤j1≤, 2r−2,
−
1, if i=1, and j =2r
118 A.LourdusamyandSherryGeorge
and
(4r 1)i 4r+8, if 2 i k, and j =1,
− − ≤ ≤
(4r 1)i 4r+4, if 2 i k, and j =2,
f(ei,j)=((44rr−−−11))ii−−−44rr++14j0,−2, iiff 22≤≤≤ii≤≤≤kk,, aanndd j4=≤3j,≤r,
(4r 1)i+4r 4j+5, if 2 i k, and r+1 j 2r 1,
(4r−−1)(i−1)−, if i=≤k,≤and j =2r. ≤ ≤ −
2, if i=1, and j =1,
3j 1, if i=1, and 2 j 4,
fv(vi,j)=484((44jr,rr−−−−−5411,j))+ii−−6,44rr++36,, iiiiifffff iii22===≤≤111ii,,,≤≤aaannnkkddd,, aarj5nn+=≤≤dd22jjjr≤≤≤==, jr12,,,≤2r−1,
(4r 1)i 4r+7, if 2 i k, and j =3,
We prove that the((((4444trrrrhe−−−−−or1111e))))miiik,−−+−is444trrrr+−+ue445wjj, h−+en47,,r =iiiiffff4i222,=≤≤≤≤nkiii=,≤≤≤≤8a.kkkn,,,daaajnnn=dddrrj4++=≤122j.r≤≤, jr,≤2r−1,
When r =4 the linear cyclic snake is a linear octagonal snake with k,k 2 cycles of C .
8
≥
Now,p=7k+1andq =8kandp+q =15k+1. Definef :E(kC ) 1,2,3, ,15k+1
n
→{ ··· }
as follows:
3j, if i=1, and 1 j 3,
≤ ≤
13, if i=1, and j =4,
f(ei,j)=37111,,655ii−−481j,2,, iiiiifffff iii22===≤111ii,,,≤aaannnkkddd,, aajj5nn==≤dd78jjj,,≤==612,,,
− ≤ ≤
11115555iiii−−−−6241,,j5,+21, iiiiffff i222=≤≤≤kiii,≤≤≤akkkn,,,daaajnnn=ddd8jj5.==≤34j,,≤7,