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An element $x$ of a finite group $G$ is said to be primary if the order of $x$ is a prime power. We define $\mathrm{csp2}(G)$ as follows: if $|x^G|$ is a prime power for every primary element $x$ of $G$, where $x^G$ is the conjugacy class of $x$ in $G$, then $\mathrm{csp2}(G)=0$; if there exists a primary element $x$ in $G$ such that $|x^G|$ is divisible by at least two distinct primes, then $\mathrm{csp2}(G)=\max \{|x^G| | x\in G\text{is primary},|x^G|\text{is divisible by at least two distinct primes}\}$. In this paper we discuss the influence of the number $\mathrm{csp2}(G)$ on the structure of $G$.

Let $R$ be a weakly factorial domain which admits at least two maximal ideals. We use ${𝒫}(R)$ to denote the set of all primary elements of $R$ and denote ${𝒫}(R)\setminus \{0\}$ by ${𝒫}{(R)}^{\ast }$. Let ${𝒬}(R)=\{Rq|q\in {𝒫}{(R)}^{\ast }\}$. We denote the subset of ${𝒬}(R)$ consisting of all $Rq\in {𝒬}(R)$ such that $q$ does not belong to the Jacobson radical of $R$ by ${𝒮}{𝒬}(R)$. With $R$, in this paper, we associate an undirected graph denoted by $ℂ𝔾\mathbb{P}(R)$ whose vertex set is ${𝒮}{𝒬}(R)$ and distinct vertices $R{q}_1$ and $R{q}_2$ are adjacent if and only if $R{q}_1+R{q}_2=R$. In Secs. 2 and 3 of this paper, we discuss results regarding the connectedness, the girth, and the clique number of $ℂ𝔾\mathbb{P}(R)$ and study the interplay between graph properties of $ℂ𝔾\mathbb{P}(R)$ and the properties of $R$. In Secs. 4 and 5 of this paper, we consider a supergraph of $ℂ𝔾\mathbb{P}(R)$, denoted by $𝕊𝔾ℂ𝔾\mathbb{P}(R)$ whose vertex set is ${𝒮}{𝒬}(R)$ and distinct vertices $R{q}_1$ and $R{q}_2$ are adjacent if and only if $R{q}_1\cap R{q}_2=R{q}_1{q}_2$ and study some graph properties of $𝕊𝔾ℂ𝔾\mathbb{P}(R)$.