On Extremal Unicyclic Molecular Graphs with Prescribed Girth and Minimal Hosoya Index

摘要

Let G be an n-vertex unicyclic molecular graph and Z(G) be its Hosoya index, let F n be the nth Fibonacci number. It is proved in this paper that if G has girth l then Z(G) ≥ F l+1+(n?l)F l +F l-1, with the equality holding if and only if G is isomorphic to $S_n^l$ , the unicyclic graph obtained by pasting the unique non-1-valent vertex of the complete bipartite graph K 1,n-l to a vertex of an l-vertex cycle C l . A direct consequence of this observation is that the minimum Hosoya index of n-vertex unicyclic graphs is 2n?2 and the unique extremal unicyclic graph is $S_n^3$ . The second minimal Hosoya index and the corresponding extremal unicyclic graphs are also determined.