Maximal Hosoya index and extremal acyclic molecular graphs without perfect matching

Let T be an acyclic graph without perfect matching and Z(T ) be its Hosoya index; let Fn be the nth Fibonacci number. It is proved in this work that Z(T ) ≤ 2F2m F2m+1 when T has order 4m with the equality holding if and only if T = T1,2m−1,2m−1, and that Z(T ) ≤ F2 2m+2 + F2m F2m+1 when T has order 4m + 2 with the equality holding if and only if T = T1,2m+1,2m−1, where m is a positive integer and T1,s,t is a graph obtained by joining an isolated vertex with an edge to the (s + 1)-th vertex (according to its natural ordering) of path Ps+t+1. c © 2005 Published by Elsevier Ltd