Unicyclic graphs possessing Kekule structures with minimal energy

Unicyclic graphs possessing Kekule structures with minimal energy are considered. Let n and l be the numbers of vertices of graph and cycle C-l contained in the graph, respectively; r and j positive integers. It is mathematically verified that for n >= 6 and l = 2r + 1 or l = 4 j + 2, S-n(4) has the minimal energy in the graphs exclusive of S-n(3), where S-n(4) is a graph obtained by attaching one pendant edge to each of any two adjacent vertices of C-4 and then by attaching n/2-3 paths of length 2 to one of the two vertices; S-n(3) is a graph obtained by attaching one pendant edge and n/2-2 paths of length 2 to one vertex of C-3. In addition, we claim that for 6 <= n <= 12, S-n(4) has the minimal energy among all the graphs considered while for n >= 14, S-n(3) has the minimal energy.