Let G be a finite group. The main supergraph S(G) is a graph with vertex set G in which two vertices x and y are adjacent if and only if o(x)|o(y) or o(y)|o(x). In an earlier paper, the main properties of this graph was obtained. The aim of this paper is to investigate the Hamiltonianity, Eulerianness and 2-connectedness of this graph. Mathematics Subject Classification (2010): 05C25, 05C50 An EP P O-group is a group that all elements have prime power order and an EP Ogroup is a group with

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... nts of prime order. Throughout this paper graph means simple graph. Suppose Γ is such a graph. The number of vertices adjacent to x is the degree of x and is denoted by deg Γ (x). If the graph Γ can not be disconnected by removing less than k vertices, then Γ is called k-connected. It is clear that every Hamiltonian graph is 2-connected. A set of all vertices in Γ such that no two of which are adjacent is an independent set for Γ. The independent number of Γ, α(Γ), is the cardinality of an independent set with maximum size. A set S of vertices of a graph Γ is a vertex cover for Γ, if every edge of Γ has at least one vertex in S as an endpoint. The vertex cover