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For a given simple graph $G$, the energy of $G$, denoted by $E(G)$, isdefined as the sum of the absolute values of all eigenvalues of its adjacencymatrix. Let $P_n^{\ell}$ be the unicyclic graph obtained by connecting a vertexof $C_\ell$ with a leaf of $P_{n-\ell}$\,. In [G. Caporossi, D. Cvetkovi\'c, I.Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Findinggraphs with extremal energy, {\it J. Chem. Inf. Comput. Sci.} {\bf 39}(1999)984--996], Caporossi et al. conjectured that the unicyclic graph with maximalenergy is $C_n$ if $n\leq 7$ and $n=9,10,11,13,15$\,, and $P_n^6$ for all othervalues of $n$. In this paper, by employing the Coulson integral formula andsome knowledge of real analysis, especially by using certain combinatorialtechnique, 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 graphwith maximal energy.

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