11 problems

Let $G$ be a non-abelian simple group. The Akbari–Ashrafi conjecture. The reduced power graph of $G$ is connected only if $G$ is isomorphic to an alternating group $A_n$. This conj…

Cameron–Ghosh conjecture. If the power graphs of $G$ and $H$ are isomorphic, then $G$ and $H$ have the same number of elements of each order.

Let $n\geq 2$ be an integer, and let $\mathcal{P}(\mathbb{Z}_{n})$ denote the power graph of the cyclic group $\mathbb{Z}_{n}$. Write $\partial_{1}^{L}$ for the relevant distance L…

Power-to-enhanced-power cograph conjecture. For every finite group $G$, if the power graph $\mathcal{P}(G)$ is a cograph, then the enhanced power graph $\mathcal{E}(G)$ is a cograp…

Let $n$ be a positive integer, and let $P(G)$ denote the power graph of a finite group $G$. Mirzargar et al.'s conjecture. The power graph $P(\mathbb{Z}_n)$ has the maximum number…

Let $G$ be a finite non-cyclic group. The proper enhanced power graph $\mathcal{P}_E^*(G)$ is the graph obtained from the enhanced power graph of $G$ by deleting the identity verte…

Let $G_1$ and $G_2$ be finite groups, and let $d4ab(G_1)$ and $d4ab(G_2)$ denote their power graphs. Power-graph order conjecture. If … then $G_1$ and $G_2$ have the same number of…

Let $n\geq 2$ be an integer, let $\mathbb{Z}_n$ be the cyclic group of order $n$, and let $\mathcal{G}(\mathbb{Z}_n)$ denote its power graph. The cyclic power-graph conjecture. The…

Let $\mathbb{Z}_n$ be the cyclic group of order $n$, and let $\mathcal{P}(\mathbb{Z}_n)$ be its undirected power graph: distinct vertices are adjacent exactly when one is a positiv…

Let $G$ be a finite group. Its exponent is the least common multiple of the orders of its elements; $G$ is a full exponent group if it contains an element whose order equals the ex…

For a finite group $G$, let $bdom? \mathcal{P}^*(G)$ denote its proper power graph, and suppose that this graph is connected. Bounded diameter conjecture. There exists a constant…