Let
R
be a finite commutative ring with unity, and let
G
=
(
V
,
E
)
be a simple graph. The zero-divisor graph, denoted by
Γ
(
R
)
is a simple graph with vertex set as
R
, and two vertices
x
,
y
∈
R
...are adjacent in
Γ
(
R
)
if and only if
x
y
=
0
. In
5
, the authors have studied the Laplacian eigenvalues of the graph
Γ
(
Z
n
)
and for distinct proper divisors
d
1
,
d
2
,
⋯
,
d
k
of
n
, they defined the sets as,
A
d
i
=
{
x
∈
Z
n
:
(
x
,
n
)
=
d
i
}
, where (
x
,
n
) denotes the greatest common divisor of
x
and
n
. In this paper, we show that the sets
A
d
i
,
1
≤
i
≤
k
are actually orbits of the group action:
A
u
t
(
Γ
(
R
)
)
×
R
⟶
R
, where
A
u
t
(
Γ
(
R
)
)
denotes the automorphism group of
Γ
(
R
)
. Our main objective is to determine new classes of threshold graphs, since these graphs play an important role in several applied areas. For a reduced ring
R
, we prove that
Γ
(
R
)
is a connected
threshold
graph if and only if
R
≅
F
q
or
R
≅
F
2
×
F
q
. We provide classes of
threshold
graphs realized by some classes of local rings. Finally, we characterize all finite commutative rings with unity of which zero-divisor graphs are not
threshold
.
Let G=(V,E) be a simple graph with vertex set as V and edge set as E. Within the framework of graph-based binary coding, a graph G is a threshold graph if and only if it can be generated by a binary ...code of the type 0s11t1…0sk1tk, where si and ti are positive integers with 1≤i≤k. This type of binary code is called a binary generating code of G. In this paper, we address a problem originally posed by Harary and Schwenk in 10 titled “Which graphs have integral spectra?” in which the main focus of our investigations lies in the Seidel Laplacian spectrum of G generated by 0s11t1…0sk1tk. We establish that all eigenvalues of G are integers and multiplicities of eigenvalues are equal to powers of bits 0 and 1 (sizes of strings of 0's and 1's) involved in a binary generating code of G. We exhibit a necessary and sufficient condition in terms of eigenvalues of the matrix associated with a graph-based G binary code to be Seidel Laplacian integral. Furthermore, for any prime p, we explore the Seidel Laplacian spectrum of zero-divisor graphs associated with the ring Zpm (where m is a positive integer) and the polynomial ring Zpx/(xp).
Let R be a finite commutative ring with unity, and let G = (V, E) be a simple graph. The zero-divisor graph, denoted by {\Gamma}(R) is a simple graph with vertex set as R, and two vertices x, y \in R ...are adjacent in {\Gamma}(R) if and only if \(xy = 0\). In 10, the authors have studied the Laplacian eigenvalues of the graph {\Gamma}(Z_n) and for distinct proper divisors d_1, d_2, \dots, d_k of n, they defined the sets as, A_{d_i} = {x \in Zn : (x, n) = d_i}, where (x, n) denotes the greatest common divisor of x and n. In this paper, we show that the sets A_{d_i}, 1 \leq i \leq k are actually orbits of the group action: Aut({\Gamma}(R)) \times R \longrightarrow R, where Aut({\Gamma}(R)) denotes the automorphism group of {\Gamma}(R). Our main objective is to determine new classes of threshold graphs, since these graphs play an important role in several applied areas. For a reduced ring R, we prove that {\Gamma}(R) is a connected threshold graph if and only if R = F_q or R = F_2 \times F_q. We provide classes of threshold graphs realized by some classes of local rings. Finally, we characterize all finite commutative rings with unity of which zero-divisor graphs are not threshold.
Let G = (V, E) be a finite simple connected graph. We say a graph G realizes a code of the type 0^s_1 1^t_1 0^s_2 1^t_2 ... 0^s_k1^t_k if and only if G can obtained from the code by some rule. Some ...classes of graphs such as threshold and chain graphs realizes a code of the above mentioned type. In this paper, we develop some computationally feasible methods to determine some interesting graph theoretical invariants. We present an efficient algorithm to determine the metric dimension of threshold and chain graphs. We compute threshold dimension and restricted threshold dimension of threshold graphs. We discuss L(2, 1)-coloring of threshold and chain graphs. In fact, for every threshold graph G, we establish a formula by which we can obtain the {\lambda}-chromatic number of G. Finally, we provide an algorithm to compute the {\lambda}-chromatic number of chain graphs.
