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

Amal S. Alali, Mohd Rashid, Asif Imtiyaz Ahmad Khan () and Muzibur Rahman Mozumder
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Amal S. Alali: Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
Mohd Rashid: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
Asif Imtiyaz Ahmad Khan: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
Muzibur Rahman Mozumder: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Mathematics, 2025, vol. 13, issue 15, 1-13

Abstract: 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.

Keywords: ring of integers modulo n; weakly zero-divisor graph; spectrum of graph; D α -matrix (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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