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We determine the groups with exactly two nonlinear non-faithful irreducible characters whose kernels intersect trivially.

Let $G$ be a finite group and write $\mathrm{cd}\left(G\right)$ for the degree set of the complex irreducible characters of $G$. The group $G$ is said to satisfy the two-prime hypothesis if for any distinct degrees $a,b\in \mathrm{cd}\left(G\right)$, the total number of (not necessarily different) primes of the greatest common divisor $gcd(a,b)$ is at most $2$. We prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to PSL${}_2\left(q\right)$ for $q\ge 7$.

Let $G$ be a finite group and construct a graph $\Delta \left(G\right)$ by taking $G\setminus \left\{1\right\}$ as the vertex set of $\Delta \left(G\right)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K\left(G\right)$ be the set consisting of the universal vertices of $\Delta \left(G\right)$ along the identity element. For a solvable group $G$, we present a necessary and sufficient condition for $K\left(G\right)$ to be nontrivial. We also develop a connection between $\Delta \left(G\right)$ and $K\left(G\right)$ when $\left|G\right|$ is divisible by two distinct primes and the diameter of $\Delta \left(G\right)$ is 2.