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 …

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL (G)= RT
(G)− RD (G), where RT (G) is the diagonal matrix of reciprocal distance degrees and RD (G) …

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 …

Let G be a simple connected graph and let S k (G) be the sum of the first k largest Laplacian
eigenvalues of G. It was conjectured by Brouwer in 2006 that S k (G)≤ e (G)+(k+ 1 2) holds …

The normalized distance Laplacian matrix of a connected graph G, denoted by D 𝓛 (G), is
defined by D 𝓛 (G)= Tr (G)− 1/2 DL (G) Tr (G)− 1/2, where D (G) is the distance matrix, the DL …

For a connected simple graph G, the generalized distance matrix is defined by D α:= α T r
(G)+(1-α) D (G), 0≤ α≤ 1, where Tr (G) is the diagonal matrix of vertex transmissions and D …

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 …

Let $ G $ be a connected graph with $ n $ vertices, $ m $ edges and having diameter $ d $.
The distance Laplacian matrix $ D^{L} $ is defined as $ D^ L= $ Diag $(Tr)-D $, where Diag …

Let $ G $ be a simple graph with order $ n $ and size $ m $. Let $ D (G)= $ diag $(d_1,
d_2,\dots, d_n) $ be its diagonal matrix, where $ d_i=\deg (v_i), $ for all $ i= 1, 2,\dots, n …

Let G be a simple graph with n vertices, m edges having Laplacian eigenvalues μ 1, μ 2,…,
μ n− 1, μ n= 0. The Laplacian energy LE (G) is defined as LE (G)=∑ i= 1 n| μ i− d¯|, where …