AbstractWe investigate eigenvalues of the zero-divisor graph $$\Gamma (R)$$
Γ
(
R
)
of finite commutative rings R and study the interplay between these eigenvalues, the ring-theoretic properties of R and the graph-theoretic properties of $$\Gamma (R)$$
Γ
(
R
)
. The graph $$\Gamma (R)$$
Γ
(
R
)
is defined as the graph with vertex set consisting of all nonzero zero-divisors of R and adjacent vertices x, y whenever $$xy = 0$$
x
y
=
0
. We provide formulas for the nullity of $$\Gamma (R)$$
Γ
(
R
)
, i.e., the multiplicity of the eigenvalue 0 of $$\Gamma (R)$$
Γ
(
R
)
. Moreover, we precisely determine the spectra of $$\Gamma ({\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p)$$
Γ
(
Z
p
×
Z
p
×
Z
p
)
and $$\Gamma ({\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p)$$
Γ
(
Z
p
×
Z
p
×
Z
p
×
Z
p
)
for a prime number p. We introduce a graph product $$\times _{\Gamma }$$
×
Γ
with the property that $$\Gamma (R) \cong \Gamma (R_1) \times _{\Gamma } \cdots \times _{\Gamma } \Gamma (R_r)$$
Γ
(
R
)
≅
Γ
(
R
1
)
×
Γ
⋯
×
Γ
Γ
(
R
r
)
whenever $$R \cong R_1 \times \cdots \times R_r.$$
R
≅
R
1
×
⋯
×
R
r
.
With this product, we find relations between the number of vertices of the zero-divisor graph $$\Gamma (R)$$
Γ
(
R
)
, the compressed zero-divisor graph, the structure of the ring R and the eigenvalues of $$\Gamma (R)$$
Γ
(
R
)
.
Planar, Outerplanar, and Toroidal Graphs of the Generalized Zero-Divisor Graph of Commutative Rings
