Maximal Hosoya index and extremal acyclic molecular graphs without perfect matching

Let TT be an acyclic graph without perfect matching and Z(T)Z(T) be its Hosoya index; let FnFn be the nth Fibonacci number. It is proved in this work that Z(T)≤2F2mF2m+1Z(T)≤2F2mF2m+1 when TT has order 4m4m with the equality holding if and only if T=T1,2m−1,2m−1T=T1,2m−1,2m−1, and that Z(T)≤F2m+22+F2mF2m+1 when TT has order 4m+24m+2 with the equality holding if and only if T=T1,2m+1,2m−1T=T1,2m+1,2m−1, where mm is a positive integer and T1,s,tT1,s,t is a graph obtained by joining an isolated vertex with an edge to the (s+1)(s+1)-th vertex (according to its natural ordering) of path Ps+t+1Ps+t+1.