Results on energies for trees with a given diameter having perfect matching

Abstract

Let Γ2nd be the set of trees with a given diameter d having a perfect matching, where 2n is the number of vertex. For a tree T in Γ2nd, let Pd+1 be a diameter of T and q = d − m, where m is the number of the edges of perfect matching in Pd+1. It can be found that the trees with minimal energy in Γ2nd for four cases q = d−2, d−3, d−4, [ d/2], and two remarks are given about the trees with minimal energy in Γ2nd for \(\tfrac{{2d - 3}} {3} \leqslant q \leqslant d - 5\) and \(\left[ {\tfrac{d} {2}} \right] + 1 \leqslant q \leqslant \tfrac{{2d - 3}} {3} - 1\).