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Local Properties
of the Total
Graph T(Γ(Zn))
Khalida Nazzal
(1)
and Manal Ghanem
(2)
(1) Department
of Mathematics,
Palestine
Technical
University-Kadoorie,
Tulkarm, West
Bank, Palestine.
Email address:
k.nazzal@ptuk.edu.ps
(2) Department of
Mathematics, Jordan
University, Amman
11942 Jordan.
Email address:
m.ghanem@ju.edu.jo
Doi :
https://doi.org/10.47013/14.2.2
Cited by :
Jordan J. Math &
Stat.,
14 (2) (2021),
221 - 230
PDF
Received on:
Jun. 14,
2019;
Accepted
on: Feb. 21,
2021
Abstract: Let R be a
commutative ring
with unity. The
total graph of
R, T (Γ(R)), is
the simple graph
with vertex set
R and two
distinct
vertices x and y
are adjacent if
x + y ϵ Z(R),
where Z(R) is
the set of all
zero divisors of
R. This paper
presents a study
of some local
properties of
the graph T (Γ(Zn)).
We answer the
question “ when
is T (Γ(Zn))
locally
connected?”. We
also prove that
the
neighborhoods of
any two distinct
vertices in T (Γ(Zn))
induce
isomorphic
graphs if and
only if n is
even.
Keywords: Total graph of
a commutative
ring, the ring
of integers
modulo n,
locally
connected graph,
locally
homogeneous
graph, local
property of a
graph,
neighborhood of
a vertex.
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