14 problems

Zero-divisor graph unimodality conjecture. The independence polynomial $I\bigl(\varGamma(\mathbb{Z}_n),x\bigr)$ is unimodal. The paper presents this as a conjecture following compu…

Let $R$ be a finite chromatic ring, and let $\Gamma(R)$ be its zero-divisor graph. Write $\omega(\Gamma(R))$ for the clique number of $\Gamma(R)$ and $\chi(\Gamma(R))$ for its chro…

Let $R$ be a finite commutative local ring with identity, of cardinality $32$, and suppose that $R$ is not a field. Let $\Gamma(R)$ be the simple graph whose vertices are the nonze…

Let $R$ be a commutative ring with non-zero unity, and let $\Gamma(R)$ be the graph whose vertices are the non-zero zero-divisors of $R$, with distinct vertices adjacent exactly wh…

Bender et al.'s conjecture. If $G(S)$ is uniquely complemented with clique number $3$ or greater and every vertex has a unique complement, then $G(S)$ is isomorphic to…

Let $S$ be a finite commutative semigroup with zero, and let $G(S)$ be its zero-divisor graph. The graph $G(S)$ is complemented if every vertex $a$ has an orthogonal vertex $b$, me…

Let $S$ be a finite commutative semigroup with zero, and let $G(S)$ be its zero-divisor graph. A graph is complemented if every vertex is orthogonal to some vertex, where adjacent…

Compressed-graph invariance conjecture. Under these hypotheses,

Compressed-graph structure conjecture. An edge joins the vertices represented by $\"]\phi(\prod_i d_i)$ and $\"]\phi(\prod_i b_i)$ if and only if

GCD-representative conjecture. For every $a\in D$,

Compressed-graph characterization conjecture. The zero-divisor graphs are isomorphic if and only if

Extension conjecture. The results obtained for quotient rings of unique factorization domains should extend to any quotient ring of a unique factorization domain.

Let $R$ be a commutative ring, and let $\Gamma(R)$ denote its zero-divisor graph. For a graph containing a cycle, write ${\rm gr}(\Gamma(R))$ for its girth. Anderson–Livingston con…

Behboodi–Rakeei conjecture. Every reduced ring with more than two minimal primes has annihilating ideal-graph of girth $3$. The paper states that this conjecture is settled by prov…