The maximum number of cycles in a graph with fixed number of edges

The main problem considered in this paper is maximizing the number of cycles in a graph with given number of edges. In 2009, Kiraly conjectured that there is constant c such that any graph with m edges has at most c(1.4)(m) cycles. In this paper, it is shown that for sufficiently large m, a graph with m edges has at most (1.443)(m) cycles. For sufficiently large m, examples of a graph with m edges and (1.37)(m) cycles are presented. For a graph with given number of vertices and edges an upper bound on the maximal number of cycles is given. Also, bounds tight up to a constant are presented for the maximum number of cycles in a multigraph with given number of edges, as well as in a multigraph with given number of vertices and edges.