Mathematics, Vol. 13, Pages 2385: On Dα-Spectrum of the Weakly Zero-Divisor Graph of Zn

Mathematics doi: 10.3390/math13152385

Authors:
Amal S. Alali
Mohd Rashid
Asif Imtiyaz Ahmad Khan
Muzibur Rahman Mozumder

Let us consider the finite commutative ring R, whose unity is 1≠0. Its weakly zero-divisor graph, represented as WΓ(R), is a basic undirected graph with two distinct vertices, c1 and c2, that are adjacent if and only if there exist r∈ ann(c1) and s∈ ann(c2) that satisfy the condition rs=0. Let D(G) be the distance matrix and Tr(G) be the diagonal matrix of the vertex transmissions in basic undirected connected graph G. The Dα matrix of graph G is defined as Dα(G)=αTr(G)+(1−α)D(G) for α∈[0,1]. This article finds the Dα spectrum for the graph WΓ(Zn) for various values of n and also shows that WΓ(Zn) for n=ϑ1ϑ2ϑ3⋯ϑtη1d1η2d2⋯ηsds(di≥2,t≥1,s≥0), where ϑi’s and ηi’s are the distinct primes, is Dα integral.

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