In this talk we will discuss how the energy (i.e., the sum of the absolute values of all eigenvalues) of so-called tadpole graphs $P_n^k$ , which are obtained by joining a vertex of a cycle $C_k$ to one of the ends of a path $P_{n-k}$, depends on $k$. Combined with earlier results [Y. Hou, I. Gutman, C.-W. Woo, Unicyclic graphs with maximal energy, Linear Algebra Appl. 356 (2002) 27-36 and E.O.D.A. Unicyclic bipartite graphs with maximum energy, MATCH Commun. Math. Comput. Chem., 66 (3) (2011) 913-926], this will be used to prove two conjectures on the largest and second-largest energy of a unicyclic graph due to Caporossi, Cvetkovi\'c, Gutman and Hansen and Gutman, Furtula and Hua respectively. Moreover, we characterise the non-bipartite unicyclic graphs whose energy is largest.