Unicyclic Graphs Possessing Kekulé Structures with Minimal Energy

摘要

Unicyclic graphs possessing Kekulé 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 $n geqslant 6$ and l = 2r + 1 or $l=4j+2, S_n^4$ 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 $6 leqslant n leqslant 12, S_n^4$ has the minimal energy among all the graphs considered while for $ngeqslant 14, S_n^3$ has the minimal energy.