For a finite group G, let $\Delta (G)$ denote the character graph built on the set of degrees of the irreducible complex characters of G. A perfect graph is a graph $\Gamma $ in which the chromatic number of every induced subgraph $\Delta $ of $\Gamma $ equals the clique number of $\Delta $ . We show that the character graph $\Delta (G)$ of a finite group G is always a perfect graph. We also prove that the chromatic number of the complement of $\Delta (G)$ is at most three.

Let $G$ be a finite group and $p$ be an odd prime. We show that if $\mathbf{O}_{p}(G)=1$ and $p^{2}$ does not divide every irreducible $p$-Brauer character degree of $G$, then $|G|_{p}$ is bounded by $p^{3}$ when $p\geqslant 5$ or $p=3$ and $\mathsf{A}_{7}$ is not involved in $G$, and by $3^{4}$ if $p=3$ and $\mathsf{A}_{7}$ is involved in $G$.

Let $p$ be a prime, $G$ a solvable group and $P$ a Sylow $p$-subgroup of $G$. We prove that $P$ is normal in $G$ if and only if $\unicode[STIX]{x1D711}(1)_{p}^{2}$ divides $|G:\ker (\unicode[STIX]{x1D711})|_{p}$ for all monomial monolithic irreducible $p$-Brauer characters $\unicode[STIX]{x1D711}$ of $G$.

For a finite group $G$, let $\unicode[STIX]{x1D6E5}(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. In this paper, we obtain a necessary and sufficient condition which guarantees that the complement of the character graph $\unicode[STIX]{x1D6E5}(G)$ of a finite group $G$ is a nonbipartite Hamiltonian graph.

Let $p$ be an odd prime. In this note, we show that a finite group $G$ is solvable if all degrees of irreducible complex characters of $G$ not divisible by $p$ are either 1 or a prime.

The prime vertex graph, $\Delta \left( X \right)$, and the common divisor graph, $\Gamma \left( X \right)$, are two graphs that have been defined on a set of positive integers $X$. Some properties of these graphs have been studied in the cases where either $X$ is the set of character degrees of a group or $X$ is the set of conjugacy class sizes of a group. In this paper, we gather some results on these graphs arising in the context of direct product of two groups.