International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 4 Issue 5, August 2020 Available Online: www.ijtsrd.com e-ISSN: 2456 – 6470
@ IJTSRD | Unique Paper ID – IJTSRD31865 | Volume – 4 | Issue – 5 | July-August 2020 Page 430
On the Radio Number of Certain Classes of Circulant Graphs
Kins Yenoke
PG & Research, Department of Mathematics, Loyola College, Chennai, Tamil Nadu, India
ABSTRACT
Radio labelling problem is a special type of assignment problem which
maximizes the number of channels in a specified bandwidth. A radio
labelling of a connected graph = (, ) is an injection ℎ: () → such
that (, ) + |() − ()| ≥ 1 + ()∀ , ∈ (), where () is the
diameter of the graph . The radio number of denoted (ℎ), is the
maximum number assigned to any vertex of . The radio number of ,
denoted (), is the minimum value of (ℎ) taken over all labelling’s ℎ of
. In this paper we have obtained the radio number certain classes of
circulant graphs, namely (; {1,2 … ⌊
2
⌋−
1}) , (; {1,
2
}) , (; {1,
3
}) and (; {1,
5
}).
KEYWORDS: Labelling, Radio labelling, Radio number, Circulant graphs
How to cite this paper: Kins Yenoke "On
the Radio Number of Certain Classes of
Circulant Graphs" Published in
International Journal
of Trend in Scientific
Research and
Development
(ijtsrd), ISSN: 2456-
6470, Volume-4 |
Issue-5, August 2020,
pp.430-434, URL:
www.ijtsrd.com/papers/ijtsrd31865.pdf
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1. INTRODUCTION
In the modern world, all the communications are based on
the wireless mode. In the end of the 20
th
century most of
the communication were developed based on the wireless
networks. These communications are working based on
the specified allocation of the electromagnetic spectrum.
One of the main applications is based on the low frequency
long wavelength waves called radio waves. Among this, a
fixed bandwidth ranges from 88.1 MHZ to 108 MHZ was
allotted for the channel assignment of FM radio stations.
88.1 MHz and finished at 108MHz This real-life minimax
problem motivated Chartrand et al. [6] in the year 2001 to
introduced a new labelling called the radio labelling. He
defined the formal graph theoretical definition for radio
labelling problem as follows:
Let () be the diameter of a connected graph G. An
injection h from the vertex set of G to N such that (, ) +
|ℎ() − ℎ()| ≥ 1 + ()) for every pair of vertices
in G. The radio number of h, denoted by (ℎ), is the
maximum number assigned to any vertex of G. The radio
number of G, denoted by (), is the minimum value of
(ℎ) taken over all labelling’s ℎ of .
The radio number problem is NP-hard [9], even for graphs
with diameter 2. In the past two decades plenty of
research articles are published in this area and also
developed new labelling problems based on the radio
number.
2. An Overview of the Paper
For the past 20 years, several authors studied the radio
labelling problem and its variations in various networks
and graphs. Radio labelling problem is a particular case of
radio K- Chromatic number [7]. In the recent years few
new labelling’s were introduced by different authors
based on the k value, namely, radio mean labelling, radio
multiplicative labelling, radial radio labelling etc. The
radio number of square cycles was determined by Liu et.al
[12]. Bharati et.al [2,3] obtained the bounds for the
hexagonal mesh as 3
2
− 3 + 2 + 12 ∑ ( − − 1) ≤
−2
=0
() ≤ (3
2
− 4 − 1) + 3 and completely determined
the radio number of graphs with small diameters.
Kchikech et.al [10] studied the radio k-labelling of graphs.
Fernandez et al. [8] computed the radio number for gear
graph. Kins et. al. [11] investigated the radio number for
mesh derived architectures and wheel extended graphs.
In this paper we have investigated the radio labelling of
certain classes of circulant graphs.
3. Circulant Graphs
Circulant graphs have been used for several decades in the
design of telecommunication networks because of their
optimal fault-tolerance and routing capabilities [5]. For
designing certain data alignment networks, the circulant
graphs are being used. for complex memory systems [13].
Most of the earlier research concentrated on using the
circulant graphs to build interconnection networks for
distributed and parallel systems [2]. By using circulant
graph we can adapt the performance of the network to
user needs. It’s a regular graph which includes standard
such as the complete graph and the cycle.
IJTSRD31865