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 ℓ n be the unicyclic graph obtained by connecting a vertex of C ℓ with a leaf of P n−ℓ . In [G. Caporossi, D. Cvetković, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, J. Chem. Inf. Comput. Sci. 39(1999) 984-996], Caporossi et al. conjectured that the unicyclic graph with maximal energy is C n if n ≤ 7 and n = 9, 10, 11, 13, 15 , and P 6 n 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 3 4 should be the unicyclic graph with maximal energy.

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 6 n 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 6 n for n = 8, 12, 14 and n ≥ 16. In"Y. Hou, I. Gutman and C. Woo, Unicyclic graphs with maximal energy, Linear Algebra Appl. 356 (2002), 27-36", the authors proved that E(P 6 n ) 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 6 n 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 6 n is greater than that of C n for n = 8, 12, 14 and n ≥ 16, which completely solves this open problem and partially solves the above conjecture.

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