On Dα-Spectrum of the Weakly Zero-Divisor Graph of ℤn - Amal S. Alali; Mohd Rashid; Asif Imtiyaz Ahmad Khan; Muzibur Rahman Mozumder (Mathematics)
Let us consider the finite commutative ring R, whose unity is <semantics>1≠0.</semantics> Its weakly zero-divisor graph, represented as <semantics>WΓ(R)</semantics>, is a basic undirected graph with two distinct vertices, <semantics>c1</semantics> and <semantics>c2</semantics>, that are adjacent if and only if there exist <semantics>r∈</semantics> <semantics>ann(c1)</semantics> and <semantics>s∈</semantics> <semantics>ann(c2)</semantics> that satisfy the condition <semantics>rs=0</semantics>. Let <semantics>D(G)</semantics> be the distance matrix and <semantics>Tr(G)</semantics> be the diagonal matrix of the vertex transmissions in basic undirected connected graph G. The <semantics>Dα</semantics> matrix of graph G is defined as <semantics>Dα(G)=αTr(G)+(1−α)D(G)</semantics> for <semantics>α∈[0,1]</semantics>. This article finds the <semantics>Dα</semantics> spectrum for the graph <semantics>WΓ(Zn)</semantics> for various values of n and also shows that <semantics>WΓ(Zn)</semantics> for <semantics>n=ϑ1ϑ2ϑ3…ϑt<msubsup>η1d1</msubsup><msubsup>η2d2</msubsup>…<msubsup>ηsds</msubsup>(di≥2,t≥1,s≥0)</semantics>, where <semantics>ϑi</semantics>’s and <semantics>ηi</semantics>’s are the distinct primes, is <semantics>Dα</semantics> integral.