The A γ matrix of a graph G is determined by A γ (G)=(1− γ) A (G)+ γ D (G), where 0≤ γ≤ 1,
A (G) and D (G) are the adjacency and the diagonal matrices of node degrees, respectively …

For a finite group G and a subset X∅ of G, the commuting graph, indicated by G= C (G, X), is
the simple connected graph with vertex set X and two distinct vertices x and y are edge …

Let R be a ring with unity. The upper ideal relation graph Γ U (R) of the ring R is the simple
undirected graph whose vertex set is the set of all non-unit elements of R and two distinct …

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 …

For a simple connected graph G of order n, the normalized Laplacian is a square matrix of
order n, defined as ℒ (G)= D (G)−<? A3B2 [thinspace]?> 1 2 L (G) D (G)−<? A3B2 …

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 …

For a finite commutative ring R with identity 1≠ 0, the weakly zero-divisor graph of R
denoted as W Γ (R) is a simple undirected graph having vertex set as a set of non-zero zero …

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 …

MOHD SHARIQ, PRAVEEN MATHIL, JITENDER KUMAR* Abstract. Let R be a commutative
ring with unity. The weakly zero-divisor graph WΓ (R) of the ring R is the simple undirected …

The normalized distance Laplacian matrix (D L-matrix) of a connected graph Γ is defined by
DL (Γ)= I− T r (Γ)− 1/2 D (Γ) T r (Γ)− 1/2, where D (Γ) is the distance matrix and T r (Γ) is the …