Unicyclic graphs with large energy

We study the energy (i.e., the sum of the absolute values of all eigenvalues) of so-called tadpole graphs, which are obtained by joining a vertex of a cycle to one of the ends of a path. By means of the Coulson integral formula and careful estimation of the resulting integrals, we prove two conjectures on the largest and second-largest energy of a unicyclic graph due to Caporossi, Cvetković, Gutman and Hansen and Gutman, Furtula and Hua, respectively. Moreover, we characterise the non-bipartite unicyclic graphs whose energy is largest.