On the center of a graph defined by Schmidt subgroups of a finite group

A Schmidt group is a non-nilpotent group whose proper subgroups are nilpotent. Schmidt graph of a finite group $G$ is the prime graph with the vertex set $\pi(G)$ in which $(p,q)$ is an edge if and only if $G$ has a Schmidt subgroup whose order is divisible by $pq$. In the paper the relationship of Schmidt graph properties with group properties is studied.