Adjacency matrix and Wiener index of zero divisor graph $$\varGamma (Z_n)$$

2020 ◽  
Author(s):  
Pradeep Singh ◽  
Vijay Kumar Bhat
Keyword(s):  
Adjacency Matrix ◽  
Zero Divisor ◽  
Wiener Index ◽  
Zero Divisor Graph

scholarly journals Distance-Based Topological Polynomials Associated with Zero-Divisor Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Ali Ahmad ◽  
S. C. López
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Wiener Index ◽  
Formula One ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Hosoya Polynomial ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity and let Z R be its set of zero divisors. The zero-divisor graph of R is the graph Γ R with vertex set V Γ R = Z R ∗ , where Z R ∗ = Z R \ 0 , and edge set E Γ R = x , y :   x ⋅ y = 0 . One of the basic results for these graphs is that Γ R is connected with diameter less than or equal to 3. In this paper, we obtain a few distance-based topological polynomials and indices of zero-divisor graph when the commutative ring is ℤ p 2 q 2 , namely, the Wiener index, the Hosoya polynomial, and the Shultz and the modified Shultz indices and polynomials.

Spectra of Boolean Graphs Over Finite Fields of Characteristic Two

2019 ◽  
Vol 63 (1) ◽  
pp. 58-65
Author(s):  
D. Scott Dillery ◽  
John D. LaGrange
Keyword(s):  
Number Theory ◽  
Finite Fields ◽  
Adjacency Matrix ◽  
Zero Divisor ◽  
Algebraic Closure ◽  
Simple Graph ◽  
Boolean Ring ◽  
Zero Divisor Graph ◽  
Finite Commutative Ring ◽  
Characteristic Two

AbstractWith entries of the adjacency matrix of a simple graph being regarded as elements of $\mathbb{F}_{2}$, it is proved that a finite commutative ring $R$ with $1\neq 0$ is a Boolean ring if and only if either $R\in \{\mathbb{F}_{2},\mathbb{F}_{2}\times \mathbb{F}_{2}\}$ or the eigenvalues (in the algebraic closure of $\mathbb{F}_{2}$) corresponding to the zero-divisor graph of $R$ are precisely the elements of $\mathbb{F}_{4}\setminus \{0\}$ . This is achieved by observing a way in which algebraic behavior in a Boolean ring is encoded within Pascal’s triangle so that computations can be carried out by appealing to classical results from number theory.

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