23 problems

Let $G$ be a finite non-commutative group, and let its commuting graph be the graph whose vertices are the elements of $G$, with two distinct vertices adjacent when they commute. I…

Let $\mathbb{F}$ be a field and let $\operatorname{Mat}_n(\mathbb{F})$ be the algebra of $n\times n$ matrices over $\mathbb{F}$. Let $\Gamma(\mathbb{F},n)$ be the graph whose verti…

Let $G$ be a finite group with trivial centre, and let its commuting graph be the graph whose vertices are the non-central elements of $G$, with two distinct vertices adjacent exac…

Let $R$ be a finite non-commutative CC-ring, meaning that all centralizers of non-central elements of $R$ are commutative, and let $\Gamma_R$ be its commuting graph. Let MSN-integr…

Let $R$ be a finite CC-ring, meaning that the centralizer of every non-central element of $R$ is commutative, and let $\Gamma_R$ be its commuting graph. A graph is CN-hyperenergeti…

Diameter-five conjecture. The commuting graph $\Gamma(\mathbb{F},n)$ has diameter equal to $5$.

Diameter-six exclusion conjecture. The diameter of a connected commuting graph $\Gamma(\mathbb{F},n)$ is never equal to $6$.

Let $G$ be a finite non-abelian group. Its commuting graph $\Gamma(G)$ is the simple graph with vertex set … where two vertices are adjacent if and only if they commute in $G$. Ira…

Let $L$ be a finite-dimensional Lie algebra over a field $\mathbb F$, and consider the process described in the paper for determining an upper bound for the diameter of its commuti…

Let $G$ be a finite solvable $A$-group, meaning that all Sylow subgroups of $G$ are abelian. Its commuting graph $\Gamma(G)$ has the noncentral elements of $G$ as vertices, with tw…

Let $G$ be a finite group with trivial center. Its commuting graph $\Gamma(G)$ has the noncentral elements of $G$ as vertices, with two distinct vertices adjacent exactly when they…

Let $n$ be a positive integer and let $F$ be a finite field. Consider the ring of $n\times n$ matrices over $F$, together with its commuting graph, whose vertices are the non-centr…

Commuting-graph recognition conjecture. The group $G$ is characterised by its commuting graph.

Let $G$ be a finite group. Its commuting graph has the elements of $G$ as vertices, with two distinct vertices adjacent when they commute. Iramanesh's conjecture. The commuting gra…

Diameter-five conjecture. The diameter of $\Gamma(\mathrm{M}_n(\mathbb{R}))$ is at most $5$.

Let $G$ be a finite group, and let $\Gamma_c(G)$ denote its commuting graph: the simple graph whose vertices are the non-central elements of $G$, with two distinct vertices adjacen…

Let $R$ be a ring with identity, let $\mathds F$ be a finite field, and let $\Gamma(R)$ denote the commuting graph of $R$, whose vertices are the non-central elements and whose edg…

Let $G$ be a finite group, and define the modified commuting graph of $G$ to have all non-trivial elements of $G$ as vertices, with two distinct vertices adjacent exactly when they…

For a finite group $G$, let $\operatorname{\mathcal{Z}}(G)$ be its center, and let $\operatorname{\Gamma}(G)$ be the graph whose vertices are the elements of…

Let $G$ be a finite group whose commuting graph is connected, and let its diameter be the greatest distance between two vertices of the graph. Common upper bound conjecture. There…

Random-group commuting graph conjecture. As $m\to\infty$, $\Gamma(G_{m,k})$ is almost surely connected and has diameter $k$.

Bounded-diameter conjecture. There is a natural number $b$ such that, whenever $\Gamma(G)$ is connected,

Let $G$ and $H$ be non-abelian finite groups, and let $Z(G)$ and $Z(H)$ denote their centers. Consider the commuting graphs on $G\setminus Z(G)$ and $H\setminus Z(H)$, whose distin…