Given a commutative ring R with identity 1≠ 0, let the set Z (R) denote the set of zero-
divisors and let Z*(R)= Z (R)\{0} be the set of non-zero zero-divisors of R. The zero-divisor …

For a finite commutative ring $ R $ with identity $1\neq 0$, the zero divisor graph $\Gamma
(R) $ is a simple connected graph having vertex set as the set of nonzero zero divisors of …

For a finite group G, the power graph P (G) is a simple connected graph whose vertex set is
the set of elements of G and two distinct vertices x and y are adjacent if and only if xi= y or …

For a finite commutative ring $\mathbb {Z} _ {n} $ with identity $1\neq 0$, the zero divisor
graph $\Gamma (\mathbb {Z} _ {n}) $ is a simple connected graph having vertex set as the …

The distance matrix D of a connected graph G is the matrix indexed by the vertices of G
where D i, j= d (vi, vj) and d (vi, vj) denotes the distance between the vertices vi and v j. The …

Let $ R $ be a finite commutative ring with non zero identity. The unit graph of $ R $ denoted
by $ G (R) $ is the graph obtained by setting all the elements of $ R $ to be the vertices of a …

For a finite commutative ring Zn with identity 1= 0, the zero divisor graph Γ (Zn) is a simple
connected graph having vertex set as the set of non-zero zero divisors, where two vertices x …

Let R be a ring with unity. The cozero-divisor graph of a ring R, denoted by Γ′(R), is an
undirected simple graph whose vertices are the set of all non-zero and non-unit elements of …

In this paper, we investigate the normalized distance Laplacian spectrum of the zero-divisor
graph of a commutative ring R, denoted by Γ (R), where R is taken as the ring of integers …

For a finite group G, the power graph P (G) is a simple connected graph whose vertex set is
the set of elements of G and two distinct vertices are adjacent if and only if one is a power of …