D3 Divisor Cordial Labelling for Butterfly Graph with Shell Order
A. Sudha Rani and S. Sindu Devi
*
SRMIST, Chennai, India
Keywords: Dihedral Group Divisor Cordial Labelling, Shell Graph, Butterfly Graph, Butterfly Graph with Shell Order.
Abstract: In this paper we introduce D3 Divisor Cordial labelling for butterfly graph with shell order, which is based
on the concept of divisor cordial labelling. For this define a function β:V(G)→D3 such that for each edge uv,
we assign the label 0 if o(u)/o(v) or o(v)/o(u) assign the label 1 if o(u) not divides o(v). Let us define the new
function µα (β) which represents several edges of G having label α under the mapping β. Now β is called D3
divisor cordial labelling if |µ1 (β) −µ0 (β)| ≤ 1. The graph which satisfies the above condition is called the D3
divisor cordial labelling graph. Here we discuss the shell graph, butterfly graph, and butterfly graph with shell
order graphs undergoing Dihedral group divisor Cordial Labelling.
1 INTRODUCTION
The butterfly graph is a graph with two vertices and
two edges, where each vertex is connected to the
other vertex by an edge, and each vertex has a loop
(an edge that connects it to itself). The shell order
refers to the placements of vertices in concentric
shells or layers around a central vertex.
The field of graph theory is very important in
many domains. One of the main applications of graph
theory is graph labelling which is utilized in several
disciplines such as database management, astronomy
etc.
Graph labelling is the process of assigning values
to vertices, edges, or both under a specific condition
or conditions. A graph labelling is a map connecting
the graph's elements to a collection of numbers,
typically a collection of nonnegative or positive
integers. Edge labelling is used when the domain is
the set of edges. Total labelling is the term used when
labels are applied to both vertices and edges.
This essay focuses on the finite, simple,
undirected graph G with p nodes and q edges. G is
also known as a (p, q) graph. For information on the
ideas of graph theory and Abstract Algebra, see the
references (Bondy and Murty 1976), (Harary 1972),
and (Dummit and Footy 2004). Gondalial 2020,
proved ring sum of the helm with star graph, gear with
star graph, double wheel with star graph, jellyfish
with star graph, and gem with star graph is a cordial
*
Corresponding author.
graph divisor for information on the ideas of divisor
cordial labelling we refer (Varatharajan et al. 2020)
introduced the divisor cordial labelling approach. We
refer to (Burton David 1980) for elementary number
theory, Pair sum labelling’s a theory that Ponraj et al
2010. Labelling is essential in areas like radar,
networks, coding theory, and signal processing. We
refer to Maya. et.al 2014 for Some New Families of
Divisor Cordial Graph. (Lawrence Rozario Raj. et.al
2014) proved “Divisor Cordial Labelling of Some
Disconnected Graphs. In this paper we discussed the
Dihedral divisor cordial labelling undergoes shell
graph S
n
, n (excluding the apex vertex), butterfly
graph, butterfly graph with shell order (q,q).
2 PRELIMINARIES
2.1 Butterfly Graph
The butterfly graph BF
m,n
is a two even cycles of the
same order say C
n
, sharing a common vertex with m
pendant edges attached at the common vertex is
called a butterfly graph.
2.2 Shell Graph
A shell S
n
is the graph obtained by taking (n-3)
concurrent chords in a cycle C
n
. The vertex at which
Rani, A. and Devi, S.
D3 Divisor Cordial Labelling for Butterfly Graph with Shell Order.
DOI: 10.5220/0012509600003739
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 1st International Conference on Artificial Intelligence for Internet of Things: Accelerating Innovation in Industry and Consumer Electronics (AI4IoT 2023), pages 429-433
ISBN: 978-989-758-661-3
Proceedings Copyright © 2024 by SCITEPRESS – Science and Technology Publications, Lda.
429
all the chords are concurrent is called the apex. The
shell is also called fan f
n−1
.
2.3 Multiple Shell Graph
A multiple shell is a collection of edge disjoint shells
that have their apex in common. Hence a double shell
consists of two edge disjoint shells with a common
apex.
2.4 Bow Graph
A bow graph is a double shell in which each shell has
any order.
2.5 Divisor Cordial Labelling
A divisor Graph G with vertex set V is cordial
labelled by a bijection f from V to {1, 2,..., | V |} such
that if each edge uv is given the label 1 if f(u) divides
f(v) or f(v) divides f(u), and 0 otherwise, then the
number of edges labelled with 0 and the number of
edges labelled with 1 differ by at most 1 then it is
called divisor cordial labelling..
3 MAIN RESULTS
3.1 D
3
Divisor Cordial Labelling
Let β: V(G) →D
3
be a mapping such that for each
edge ab we assign the label 0 if order of u divides
order of v or order of v divides order of u and assign
the label 1 if order of u not divides order of v. Let us
define the new function
which representthe
number of edges of G having label α under the
mapping β. Now β is called Dihedral group divisor
cordial labelling if
and
where
represents number
of vertices having label a under .The graph which
satisfies above condition is called Dihedral group
divisor cordial labelling graph.
3.2 Order of an Element
Let G be an undirected graph without loops and
multiple edges. Let us consider Dihedral group whose
elements are e, a,
,b,ab,
b whose structure is given
below e = (1)(2)(3), a = (123),
= (132), b=(12), ab
= (13),
b = (23) Order of each element is given by
O(e) =1,O(a)=3,O(
)=3,O(b)=2,
O(ab)=2,O(
b)=2.
In this section we discuss the shell graph, butterfly
graph and butterfly graph with shell order graphs
undergoes Dihedral group divisor Cordial Labelling.
Theorem: 3.1
The shell graph S
n
, n (excluding the apex vertex)
is D
3
divisor cordial labelling graph.
Proof:
G should be a shell graph. The vertices and edges of
the graph G are defined as V(G) ={k, m
j
: j varies is
from 1 to n and E(G) ={
= k m
j
: j varies is from 1 to
n;
= m
j
:1 ≤ j ≤ (n − 1)}
For n ≥ 6 we discuss six cases, consider the apex
vertex k = a
Case 1:
Let
Let us define the function as β
by assigning as
,
,
,
,
,
.
In this instance the vertices labelled as e,
,
b,
, ab will appears p times and the vertex a will
appear 2p times in the D
3
Dihedral group, and each
edge labelled as 0 will appear 6p-1 times and 1will
occur 6p times respectively. As a result, in this
instance we obtain β as Dihedral group D
3
divisor
cordial labelling.
Case 2:
Let
we assign the same labelling as in case 1,
For thr remaining vertices we assigning
as
In this instance, the vertices e,
, b.
will
appears p times and the vertices a, ab will appear 2p
times. in the D
3
Dihedral group, and each edge
labelled as 0 will appear 6p times and 1will occur
6p+1 times respectively. As a result, in this instance
we obtain β as Dihedral group D
3
divisor cordial
labelling.
Case 3:
Let
we assign the same labelling as in case 1
For the remaining vertices we assigning as
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430
In this instance, the vertices a,
,
will appears
2p times and the vertices e, b, ab will appear p times
in the D
3
Dihedral group, and each edge labelled as
0will appear 6p+1 times and 1 will occur 6p+2 times
respectively. As a result, in this instance we obtain β
as Dihedral group D
3
divisor cordial labelling.
Case 4:
Let
we assign the same labelling as in case 1
For the remaining vertices, we assign as
In this instance, the vertices e, a,
, b will appear
2p times and the vertices ab,
will appear p times
in the D
3
Dihedral group, and each edge labelled as 0
will appear 6p+3 times and 1 will occur 6p+2 times
respectively. As a result, in this instance we obtain β
as Dihedral group D
3
divisor cordial labelling.
Case 5:
Let
we assign the same labelling as in case 1
For the remaining vertices we assigning as
,
,
,
In this instance, the vertices e,
, a.
,ab will
appear 2p times and the vertex b will appear p times
in the D
3
Dihedral group, and each edge labelled as
0will appear 6p+4 times and 1 will occur6p+3 times
respectively. As a result, in this instance we obtain
β as Dihedral group D
3
divisor cordial labelling.
Case 6:
Let
we assign the same labelling as in case 1
For the remaining vertices we assigning to
,
,
,
,
,
In this instance, all the vertices e,
, a., ab, b
will appear 2p times in the D
3
Dihedral group, and
each edge labelled as 0will appear 6p+5 times and
1will occur 6p+4 times respectively. As a result, in
this instance we obtain β as Dihedral group D
3
divisor
cordial labelling.
Table 1.
Nature
P
2p
P
P
2P
p
p
2P
2p
2p
2p
2p
2p
2p
2p
2p
2p
2p
Table 1 shows number of times the vertices
,
,
will appear.
Table 2.
P
P
P
6p
6p-1
2P
P
P
6p+1
6p
p
P
2p
6p+2
6p+1
P
2p
P
6p+2
6p+3
2p
P
2p
6p+3
6p+4
2p
2p
2p
6p+4
6p+5
Table 2 shows number of times the vertices
,
,
will appear and also shows
the number of times edges labelled as 0 and 1.
Hence all butterfly graphs
are Dihedral
group divisor cordial labelling graph.
Theorem 3.2
The butterfly graph BF
q,q
is D
3
divisor cordial
labelling for all n ≥ 3.
Proof: Let
is a butterfly graph whose
vertices are given by
,
and
edge set is E (
we discuss three cases, we consider
as
apex vertex for all the remaining cases.
Case 1: I
Let
Let us define the function as β
by
assigning as
D3 Divisor Cordial Labelling for Butterfly Graph with Shell Order
431
In this instance, all the vertices e, a, b, ab,
will
appear p times, the vertex
will appear 2p timesin
the D
3
Dihedral group and each edge labelled as 0 will
appear 4p+1 times and 1will occur 4p+1 times
respectively. As a result, in this instance we obtain β
as Dihedral group D
3
divisor cordial labelling.
Case 2:
Let .
we assign the same labelling as in case 1,
For the remaining vertices we assigning to
In this instance, the vertices e,
,b will appear 2p
times and a, ab,
will appear p timesin the D
3
Dihedral group, and each edge labelled as 0 will
appear 4p+3 times and 1 will occur4p+3 times
respectively. As a result, in this instance we obtain β
as Dihedral group D
3
divisor cordial labelling.
Case 3:
Let
we assign the same labelling as in case 1,
For the remaining vertices we assigning as
In this instance, the vertex a will appear p times
and
, b,
,e,ab will appear 2p timesin the D
3
Dihedral group, and each edge labelled as 0 will
appear 4p+5 times and 1 will occur 4p+5 times
respectively. As a result, in this instance we obtain β
as Dihedral group D
3
divisor cordial labelling.
Table 3.
Nature
P
P
2p
P
2p
P
2p
2p
2p
P
2p
2p
Table 3 shows number of times the vertices
,
,
,
will appear.
Table 4.
P
P
4p+1
4p+1
P
P
4p+3
4p+3
2p
2p
4p+5
4p+5
Table 4 shows number of times the vertices
,
will appear and number of times the
edges labelled 0 and 1.
Hence all butterfly graphs
are Dihedral
group divisor cordial labelling graph.
Theorem 3.3
The butterfly graph with shell order (q, q) (order
excludes the apex) is Dihedral group divisor cordial
labelling.
Proof:-
Make G a butterfly graph by omitting the apex
from the shell of order (q,q). Define V(G) = {k
0
, k
1
,
k
2
, m
j
, n
j
: 1 ≤ j ≤ y} and E(G) =
= k
0
k
1
,
= k
0
k
2
,
e
j
= k
0
m
i
and e
2y−1+j
= k
0
n
j
: 1 ≤ j ≤ y, e
y+j
= m
j
m
j+1
and
e
3y−1+j
= n
j
n
j+1
: 1 ≤ j ≤ (y − 1)}are the vertices and
edges of the graph G.
Here For n ≥ 3 we discuss 3 cases by keeping k
0
= a, k
1
= b, k
2
= e as apex vertices.
Case 1:
Let n = 3p, p ≥ 1,.
In this instance, the vertices labelled as b, a, e will
appear 2p times, the vertices labelled as
, ab,
will appear p times in the D
3
Dihedral group, and each
edge labelled as 0 appearing 6p times and 1 appearing
6p times respectively. As a result, in this instance, we
obtain β as Dihedral group D
3
divisor cordial
labelling.
Case 2:
Let n = 3p+1, p ≥ 1,. For 1 ≤ j ≤ 3p we assign
the same labelling as in case 1
The remaining vertices are labelled as
In this instance, the vertices labelled as e, a,
,b
will appear 2p times, the vertices labelled as ab,
will appear p times in the D
3
Dihedral group, and each
edge labelled as 0 appearing 6p+2 times and 1
appearing 6p+2 times respectively. As a result, in this
instance, we obtain β as Dihedral group D
3
divisor
cordial labelling.
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432
Case 3:
Let n = 3p+2, p ≥ 1,.
For 1 ≤ j ≤ 3p we assign the same labelling as in
case 1. The remaining vertices are labelled as
In this instance, the vertices labelled as b will
appear 3p times, the vertices labelled as e, a,
, ab,
will appear 2p times in the D
3
Dihedral group
each edge labelled as 0 appearing 6p+4 times and 1
appearing 6p+4 times respectively. As a result, in this
instance, we obtain β as Dihedral group D
3
divisor
cordial labelling.
Hence the butterfly graph with shell order (q, q) is
Dihedral group D
3
divisor cordial labelling.
4 CONCLUSIONS
In this paper, we show that butterfly graph, butterfly
with the shell order (q,q), and butterfly with the shell
order (q, 2q) undergoes Dihedral group D
3
divisor
cordial labelling In the future, we intend to exhibit
numerous graph labels of graphs connected to shells.
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