Ordering of unicyclic graphs by minimal energies and Hosoya indices

摘要

Ordering of the unicyclic graphs with n vertices according to their minimal energies is investigated by employing the method of coefficient comparison, the Coulson integral formula, the theorem of zero points, and the relation between the roots and the coefficients of a polynomial equation. We derive the first 16 unicyclic graphs for n >= 1784. We list the first 17, 18, 19, 20, 16, 17, 15, 14, 13, and 9 unicyclic graphs for 1783 >= n >= 1751, 1750 >= n >= 76, 75 >= n >= 71, 70 >= n >= 58, 57 >= n >= 13, n = 12, n = 11,10, n = 9, n = 8, and n = 7, respectively. In addition, we obtain the first 12 unicyclic graphs with minimal Hosoya indices in the increasing order for n >= 17.