Graphs associated to conjugacy classes of normal subgroups in finite groups

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We attach to $N$ two graphs ${\mathrm{\Gamma }}_G(N)$ and ${\mathrm{\Gamma }}_G^{\ast }(N)$ related to the conjugacy classes of $G$ contained in $N$ and to the set of primes dividing the sizes of these classes, respectively. These graphs are subgraphs of the ordinary ones associated to the conjugacy classes ofG, $\mathrm{\Gamma }(G)$ and ${\mathrm{\Gamma }}^{\ast }(G)$, which have been widely studied by several authors. We prove that the number of connected components of both graphs is at most 2, we determine the best upper bounds for the diameters and characterize the structure of $N$ when these graphs are disconnected.