The energy of a graph G is defined as the sum of the absolute values of all the eigenvalues of the graph. Let U(k) denote the set of all unicyclic graphs of order 2k which have a perfect matching. S~1_3(k) denotes the unicyclic graph on 2k vertices obtained from a triangle C3 by attaching one pendant edge and k-2 paths of length 2 together to one of the vertices of C _3, and S~1_4(k) denotes the unicyclic graph on 2k vertices obtained from a cycle C4 by attaching one path P of length 2 to one of the four vertices of C4 and then attaching k -3 paths of length 2 to the middle vertex of the path P. In the paper "X. Li, J. Zhang and B. Zhou, On unicyclic conjugated molecules with minimal energies, J. Math. Chem. 42 (2007) 729-740", the authors proved that either S_3~1/(k) or S_4~1(k) is the graph with the minimal energy in U(k). They remarked that computation result shows that the energy of S_3~1(k) is greater than that of S_4~1(k) for larger k. However they could not find a proper way to prove this, and finally they conjectured that S_4~1(k) is the unique graph with minimal energy in U(k). This short note is to give a confirmative proof to the conjecture.
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