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Dupont, Miriam Rodr\\'iguez","submitted_at":"2017-07-04T18:29:10Z","abstract_excerpt":"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. 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