On D α -Spectrum of the Weakly Zero-Divisor Graph of Z n

Let us consider the finite commutative ring R , whose unity is 1 ≠ 0 . Its weakly zero-divisor graph, represented as W Γ ( R ) , is a basic undirected graph with two distinct vertices, c 1 and c 2 , that are adjacent if and only if there exist r ∈ ann ( c 1 ) and s ∈ ann ( c 2 ) that satisfy the condition r s = 0 . Let D ( G ) be the distance matrix and T r ( G ) be the diagonal matrix of the vertex transmissions in basic undirected connected graph G . The D α matrix of graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) for α ∈ [ 0 , 1 ] . This article finds the D α spectrum for the graph W Γ ( Z n ) for various values of n and also shows that W Γ ( Z n ) for n = ϑ 1 ϑ 2 ϑ 3 ⋯ ϑ t η 1 d 1 η 2 d 2 ⋯ η s d s ( d i ≥ 2 , t ≥ 1 , s ≥ 0 ) , where ϑ i ’s and η i ’s are the distinct primes, is D α integral.