Maximal Hosoya index and extremal acyclic molecular graphs without perfect matching

AbstractLet 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)≤2F2mF2m+1 when T has order 4m with the equality holding if and only if T=T1,2m−1,2m−1, and that Z(T)≤F2m+22+F2mF2m+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

On acyclic conjugated molecules with minimal energies

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. In [5] Gutman (J. Math. Chem.1 (1987) 123-143) proposes two conjectures about the minimum of the energy of conjugated trees (trees with a perfect matching). This paper mathematically verifies the two conjectures. In addition, trees with the second and the third smallest energies are also discussed. (C) 1999 Published by Elsevier Science B.V. All rights reserved