  For a finite group $G$ with a normal subgroup $H$, the enhanced quotient
graph of $G/H$, denoted by $\mathcal{G}_{H}(G),$ is the graph with vertex set
$V=(G\backslash H)\cup \{e\}$ and two vertices $x$ and $y$ are edge connected
if $xH = yH$ or $xH,yH\in \langle zH\rangle$ for some $z\in G$. In this
article, we characterize the enhanced quotient graph of $G/H$. The graph
$\mathcal{G}_{H}(G)$ is complete if and only if $G/H$ is cyclic, and
$\mathcal{G}_{H}(G)$ is Eulerian if and only if $|G/H|$ is odd. We show some
relation between the graph $\mathcal{G}_{H}(G)$ and the enhanced power graph
$\mathcal{G}(G/H)$ that was introduced by Sudip Bera and A.K. Bhuniya (2016).
The graph $\mathcal{G}_H(G)$ is complete if and only if $G/H$ is cyclic if and
only if $\mathcal{G}(G/H)$ is complete. The graph $\mathcal{G}_H(G)$ is
Eulerian if and only if $|G|$ is odd if and only if $\mathcal{G}(G)$ is
Eulerian, i.e., the property of being Eulerian does not depend on the normal
subgroup $H$.
