Available via license: CC BY 3.0
Content may be subject to copyright.
Journal of Physics: Conference Series
PAPER • OPEN ACCESS
Root square mean labelling of some graphs obtained from path
To cite this article: R Ramdani et al 2021 J. Phys.: Conf. Ser. 1869 012143
View the article online for updates and enhancements.
This content was downloaded from IP address 158.46.160.221 on 16/04/2021 at 17:01
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution
of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Published under licence by IOP Publishing Ltd
Annual Conference on Science and Technology (ANCOSET 2020)
Journal of Physics: Conference Series 1869 (2021) 012143
IOP Publishing
doi:10.1088/1742-6596/1869/1/012143
1
Root square mean labelling of some graphs obtained from
path
R Ramdani1, *, A Y Fanisa1, S Gumilar2 and S Sarbini3
1 Department of Mathematics, Faculty of Science and Technology, UIN Sunan
Gunung Djati Bandung, Indonesia
2 Jurusan Sejarah Kebudayaan Islam, Fakultas Adab Dan Humaniora, UIN Sunan
Gunung Djati Bandung, Indonesia
3 Jurusan Psikologi, Fakultas Psikologi, UIN Sunan Gunung Djati Bandung, Indonesia
*rismawatiramdani@uinsgd.ac.id
Abstract. Let G be a graph with q edges. A labelling f of G is said to be root square mean
labelling if such that when each edge is labelled with
or
, then the resulting edge labels are distinct. A
graph G is called a root square mean graph if G can be labelled by a root square mean labelling.
In this paper we determine a root mean square labelling of two graphs obtained from path, which
are corona product of ladder and complete graph with order 1, and a graph obtained from
triangular snake by join one vertex with degree 2 in each triangle to a new vertex. The method
of labelling construction is we need to do labeling to the vertices of the graph with label 1, 2, 3,
…, q+1. The labels of the vertices are not necessarily different. The next step is we need to do
labeling to the edges with the certain formula by using the vertex labeling. The edge labels must
be different. By the labelling we construct, we proof that the two graphs are root square mean
graphs.
1. Introduction
Root square mean labelling was introduced by Sandhya et al. [1]. A labelling of a graph with
edges is said to be root square mean labelling if such that when each
edge is labeled with
or
, then the resulting edge
labels are distinct. A graph is called a root square mean graph if it is possible to label the vertices of
by root square mean labeling. In the same paper, they prove that path, cycle, comb, ladder, triangular
snake, quadrilateral snake, star, and complete graph, are root square mean graphs. Other results about
root square mean labelling was given by Sandhya, Somasundaram, and Anusa [2].Sandhya continue the
before research about root square mean graph and gave some new disconnected root square mean graphs,
one of them is . Sandhya et al., gave some root square mean graphs, among others
are kite graphs, crown graphs, and a graph obtained by attaching a pendent edge to both sides of each
vertex of a path Given root square mean labelling of new crown graphs [5].
Another variation of root square mean graph is super-root square mean labelling. This labelling was
introduce by Thirugnanasambandam and Venkatesan [6].
Annual Conference on Science and Technology (ANCOSET 2020)
Journal of Physics: Conference Series 1869 (2021) 012143
IOP Publishing
doi:10.1088/1742-6596/1869/1/012143
2
Let be a graph with vertices and edges. Let be an injective function.
For a vertex labelling f, the induced edge labelling is defined by
or
then f is called a super root square mean if
. A graph which admits super root square mean labelling is called
super root square mean graph. In this paper, we investigate super root square mean labelling of some
graphs.
Gopi proved that the slanting ladder is a super root square mean graph [7]. Sandhya proved that
double triangular snake, alternate double triangular snake, double quadrilateral snake and alternate
double quadrilateral snake graphs are super root square mean graphs [8]. Devi and Kumar proved that
m copies of path , some copies of complete graph, corona product of path and complement of complete
graph , middle graph of path, and dragon graph are super root square mean graphs [9].
Gowri and Vembarasi introduced root cube mean labelling of graphs [10]. A graph with
vertices and edges is said to be a root cube mean graph if it is possible to label the vertices
with distinct elements from in such a way that when each edge is labelled
with
or
, then the resulting edge labels are distinct.
Here is called a root cube mean labelling of .
Next, we will write the definition of some graphs used in this research.
Path with order , denoted by is a graph with vertices of degree and vertices of degree .
Complete graph with order , denoted by is a graph with the two vertices are adjacent. Complete
graph with order 1, denoted by , is a graph with one vertex and has no edge.
Let be a graph with order and be a graph with order . The corona product is obtained
by taking one copy of and copies of ; and by joining each vertex of the -th copy of to the -th
vertex of , where .
The Cartesian product of graphs and is a graph such that the vertex set of is the
Cartesian product and any two vertices and are adjacent in if and
only if either and is adjacent with in , or and is adjacent with in .
Ladder is the Cartesian product of path with order and path with order .
A Triangular Snake is obtained from a path by joining and to a new vertex for
. That is every edge of a path is replaced by a triangle
Aims of this research are as follows:
To prove that the corona product of ladder and complete graph with order is a root square
mean graphs.
To proof that the graph obtained from triangular snake by join one vertex with degree in each
triangle to a new vertex is a root square mean graphs.
2. Methodology
The method we use in this research are literature study and analyzing. In the literature study, we check
some research about this topic. Then, we continue the research. We choose the graphs for analyzing.
Then, we construct a root square mean labeling of the graphs and proof that the graphs are root square
mean graphs. To proof that a graph is a root square mean graph, we need to construct a root square mean
labeling of the graph. The first step of the labeling is we need to do labeling to the vertices of the graph
with label 1, 2 , 3, …, q+1, where q is the number of edges of the graph. The labels of the vertices are
not necessary different. The next step is we need to do labeling to the edges with the certain formula by
using the vertex labeling. The edge labels must be different.
Annual Conference on Science and Technology (ANCOSET 2020)
Journal of Physics: Conference Series 1869 (2021) 012143
IOP Publishing
doi:10.1088/1742-6596/1869/1/012143
3
3. Results and discussion
In this research, we determine root square mean labelling that introduced by Sandhya in [1]. We use
the similar method with proof method [1-4], for some different graphs. We construct the original
formula of root square mean labelling of other graphs.
In the first theorem, we will proof that the corona product of ladder and complete graph with order
is a root square mean graphs.
Theorem 1. If denote ladder with order and denote complete graph with order , then
is a root square mean graphs.
Proof. denote ladder with order and denote complete graph with one vertex. Let the vertex
set of is
and the edge set of is
An illustration of can be seen in the Figure 1.
Figure 1. An illustration of .
From the definition of , we can see that has edges.
Define a labelling from to as follows:
From the vertex labelling, we have the edge labelling as follows:
Annual Conference on Science and Technology (ANCOSET 2020)
Journal of Physics: Conference Series 1869 (2021) 012143
IOP Publishing
doi:10.1088/1742-6596/1869/1/012143
4
for .
It can be seen that there are no two edges with the same label. So that, is a root square mean labeling
of . Futhermore, we can conclude that is a root square mean graph.
In the second theorem, we will proof that the graphs obtained from path by joining
and to a new vertex and joining to new vertex , for , is a root square mean
graph.
Theorem 2. Let G be a graph graphs obtained from path by joining and to a new
vertex and joining to new vertex , for . Then G is a root square mean graph.
Proof. An illustration of can be seen in Figure 2.
Figure 2. An illustration of G.
From the illustration in the Figure 2, we know that the number of edges of is .
Define a labelling from to as follows:
From the vertex labelling, we have the edge labelling as follows:
for , also
Annual Conference on Science and Technology (ANCOSET 2020)
Journal of Physics: Conference Series 1869 (2021) 012143
IOP Publishing
doi:10.1088/1742-6596/1869/1/012143
5
for .
It can be seen that there are no two edges with the same label. So that, is a root square mean labelling
of . Futhermore, we can conclude that is a root square mean graph.
4. Conclusion
From Theorem 1, we can see that there are a root square mean labeling of the corona product of ladder
and complete graph with order . Also from Theorem 2, we can check that we can construct a root
square mean labeling of the graphs obtained from path by joining and to a new vertex
and joining to new vertex , for . So that, we can conclude that the the corona
product of ladder and complete graph with order and the graphs obtained from path by
joining and to a new vertex and joining to new vertex , for are root
square mean graphs.
Acknowledgement
Author acknowledges the support by UIN Sunan Gunung Djati Bandung. The author wish to thank the
referees for their thoughtful suggestions.
References
[1] Sandhya S S, Somasundaram S and Anusa S 2014 Root square mean labeling of graphs Int. J.
Contemp. Math. Sci.
[2] Sandhya S ., Somasundaram S and Anusa S 2014 Root Square Mean Labeling of Some New
Disconnected Graphs Int. J. Math. Trends Technol.
[3] Sandhya S S, Somasundaram S and Anusa S 2015 Root Square mean labeling of some more
disconnected graphs Int. Math. Forum
[4] Sandhya S S, Somasundaram S and Anusa S 2015 Some More Results on Root Square Mean
Graphs J. Math. Res.
[5] Anon 2020 Root Square Mean Labeling (RSML) of New Crown Graphs Int. J. Recent Technol.
Eng.
[6] Ponraj R and Somasundaram S 2008 Mean labeling of graphs obtained by identifying two graphs
J. Discret. Math. Sci. Cryptogr.
[7] Gopi R 2017 Super Root Square Mean Labeling of Some Graphs Int. J. Eng. Sci. Adv. Comput.
Bio-Technology
[8] Sandhya S S, Somasundaram S and Anusa S 2016 Some Results on Super Root Square Mean
Labeling Int. J. Math. Soft Comput.
[9] devi R C and Kumar S S 2016 Super Root Square Mean Labeling of Some Graphs Int. J. Math.
Trends Technol.
[10] Gowri R and Vembarasi G 2017 Root Cube Mean Labeling of Graphs Int. J. Eng. Sci. Adv.
Comput. Bio-Technology
