  The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum of
the absolute values of all eigenvalues of its adjacency matrix. Denote by $C_n$
the cycle, and $P_n^{6}$ the unicyclic graph obtained by connecting a vertex of
$C_6$ with a leaf of $P_{n-6}$\,. Caporossi et al. conjecture that the
unicyclic graph with maximal energy is $P_n^6$ for $n=8,12,14$ and $n\geq 16$.
In``Y. Hou, I. Gutman and C. Woo, Unicyclic graphs with maximal energy, {\it
Linear Algebra Appl.} {\bf 356}(2002), 27--36", the authors proved that
$E(P_n^6)$ is maximal within the class of the unicyclic bipartite $n$-vertex
graphs differing from $C_n$\,. And they also claimed that the energy of $C_n$
and $P_n^6$ is quasi-order incomparable and left this as an open problem. In
this paper, by utilizing the Coulson integral formula and some knowledge of
real analysis, especially by employing certain combinatorial techniques, we
show that the energy of $P_n^6$ is greater than that of $C_n$ for $n=8,12,14$
and $n\geq 16$, which completely solves this open problem and partially solves
the above conjecture.
