Complete solution to a conjecture on the maximal energy of unicyclic graphs

For a given simple graph $G$, the energy of $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let $P_n^{\ell}$ be the unicyclic graph obtained by connecting a vertex of $C_\ell$ with a leaf of $P_{n-\ell}$\,. In [G. Caporossi, D. Cvetković, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, {\it J. Chem. Inf. Comput. Sci.} {\bf 39}(1999) 984--996], Caporossi et al. conjectured that the unicyclic graph with maximal energy is $C_n$ if $n\leq 7$ and $n=9,10,11,13,15$\,, and $P_n^6$ for all other values of $n$. In this paper, by employing the Coulson integral formula and some knowledge of real analysis, especially by using certain combinatorial technique, we completely solve this conjecture. However, it turns out that for $n=4$ the conjecture is not true, and $P_4^3$ should be the unicyclic graph with maximal energy.