Unicyclic Graphs Possessing Kekulé Structures with Minimal Energy

Abstract

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 $$n \geqslant 6$$ 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 $$n\geqslant 14, S_n^3$$ has the minimal energy.