2017/07/04 by Dupont, Luis A., Mendoza, Daniel G., Rodríguez, Miriam
#05C25 #05C38 #05C45 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1707.01127
For a finite group G with a normal subgroup H, the enhanced quotient graph of G/H, denoted by GH(G), is the graph with vertex set V=(G\backslash H)∪ \e\ and two vertices x and y are edge connected if xH = yH or xH,yH∈ ⟨ zH⟩ for some z∈ G. In this article, we characterize the enhanced quotient graph of G/H. The graph GH(G) is complete if and only if G/H is cyclic, and GH(G) is Eulerian if and only if |G/H| is odd. We show some relation between the graph GH(G) and the enhanced power graph G(G/H) that was introduced by Sudip Bera and A.K. Bhuniya (2016). The graph GH(G) is complete if and only if G/H is cyclic if and only if G(G/H) is complete. The graph GH(G) is Eulerian if and only if |G| is odd if and only if G(G) is Eulerian, i.e., the property of being Eulerian does not depend on the normal subgroup H.