Cut edges and Central vertices of zero divisor graph of the ring of integers modulo n
Nabajit Talukdar
TL;DR
This work studies the zero-divisor graph $Γ(ℤ_n)$, aiming to classify cut-edges and central vertices. The authors leverage the congruence condition $a x ≡ 0 (mod n)$ and the gcd structure to derive that a cut-edge $(a,b)$ occurs iff $gcd(a,n)=2$, $gcd(b,n)≥3$ and $2b=n$ (with no cut-edges when $n$ is odd). They also provide a general description of the central vertices as $ann(p)$ for primes $p|n$, with explicit forms for the prime-power and semiprime cases: for $n=p^k$ the centers are $\{p^{k-1}, 2p^{k-1}, ..., (p-1)p^{k-1}\}$, and for $n=pq$ the centers are the two arithmetic progressions $\{p,2p,...,(q-1)p\}$ and $\{q,2q,...,(p-1)q\}$. The results imply a complete bipartite structure in the semiprime case and diameter $diam(Γ(ℤ_n))=3$ when $n>pq$.
Abstract
The zero divisor graph of a commutative ring $R$ with unity is a graph whose vertices are the nonzero zero-divisors of the ring, with two distinct vertices being adjacent if their product is zero. This graph is denoted by $Γ(R)$. In this article we determine the cut-edges and central vertices in the graph $Γ(\mathbb{Z}_{n})$.