Affiliation:
1. Department of Mathematics , Aligarh Muslim University , Aligarh , India
Abstract
Abstract
Let 𝑅 be a commutative ring with identity
1
≠
0
1\neq 0
and let
Z
(
R
)
′
Z(R)^{\prime}
be the set of all non-zero and non-unit elements of ring 𝑅.
Further,
Γ
′
(
R
)
\Gamma^{\prime}(R)
denotes the cozero-divisor graph of 𝑅, is an undirected graph with vertex set
Z
(
R
)
′
Z(R)^{\prime}
, and
w
∉
z
R
w\notin zR
and
z
∉
w
R
z\notin wR
if and only if two distinct vertices 𝑤 and 𝑧 are adjacent, where
q
R
qR
is the ideal generated by the element 𝑞 in 𝑅.
In this paper, we find the signless Laplacian eigenvalues of the graphs
Γ
′
(
Z
n
)
\Gamma^{\prime}(\mathbb{Z}_{n})
for
n
=
p
1
N
p
2
p
3
n=p_{1}^{N}p_{2}p_{3}
and
p
1
N
p
2
M
p
3
p_{1}^{N}p_{2}^{M}p_{3}
, where
p
1
,
p
2
,
p
3
p_{1},p_{2},p_{3}
are distinct primes and
N
,
M
N,M
are positive integers.
We also show that the cozero-divisor graph
Γ
′
(
Z
p
1
p
2
)
\Gamma^{\prime}(\mathbb{Z}_{p_{1}p_{2}})
is a signless Laplacian integral.
Funder
Science and Engineering Research Board
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