On extremal unicyclic molecular graphs with maximal Hosoya index

Let GG be a unicyclic nn-vertex graph and Z(G)Z(G) be its Hosoya index, let FnFn stand for the nnth Fibonacci number. It is proved in this paper that Z(G)≤Fn+1+Fn−1Z(G)≤Fn+1+Fn−1 with the equality holding if and only if GG is isomorphic to CnCn, the nn-vertex cycle, and that if G≠CnG≠Cn then Z(G)≤Fn+1+2Fn−3Z(G)≤Fn+1+2Fn−3 with the equality holding if and only if G=QnG=Qn or DnDn, where graph QnQn is obtained by pasting one endpoint of a 3-vertex path to a vertex of Cn−2Cn−2 and DnDn is obtained by pasting one endpoint of an (n−3)(n−3)-vertex path to a vertex of C4C4.