On extremal unicyclic molecular graphs with maximal Hosoya index ?
Let G G be a unicyclic n n -vertex graph and Z(G) Z ( G ) be its Hosoya index, let Fn F n stand for the n n th Fibonacci number. It is proved in this paper that Z(G)?Fn+1+Fn?1 Z ( G ) ? F n + 1 + F n ? 1 with the equality holding if and only if G G is isomorphic to Cn C n , the n n -vertex cycle, and that if G?Cn G ? C n then Z(G)?Fn+1+2Fn?3 Z ( G ) ? F n + 1 + 2 F n ? 3 with the equality holding if and only if G=Qn G = Q n or Dn D n , where graph Qn Q n is obtained by pasting one endpoint of a 3-vertex path to a vertex of Cn?2 C n ? 2 and Dn D n is obtained by pasting one endpoint of an (n?3) ( n ? 3 ) -vertex path to a vertex of C4 C 4 .
unicyclic molecular graph hosoya index matching