On The Spectral Radius Of Weighted Trees And Halin Graph

The estimate of the bound of the radius is a topic in spectral theory of graphs. There are already many theories and techniques in this field. We usually study the spectrum of simple connected graphs, achieving some new results. In this paper, we mainly discuss the second largest spectrum of weighted trees and Halin graph. The analysis uses the theory of nonnegative matrices and applies the "moving edge" technique. Some results will be given in this thesis.1. Let T_n^w be any weighted tree of order n with weight w₁≥ w₂≥ ... ≥ w_n-1 > 0, and T_n^w(?) K_1,n-1^w Then ρ(T_n^w) < ρ(S_n-3,1^w(w*)(See Chapter 2 Fig.2).2. Let T_n^w be any weighted tree of order n with weight w₁≥ w₂≥ ... ≥ w_n-1 > 0, and T_n^w(?) K_1,n-1^w , T_n^w(?) (S_n-3,1^w(w*). Then(1) ρ(T_n^w) <ρ(S_n-3,1^w(w*1), (See Chapter 2 FigA), if w _n-1²(2)ρ(T_n^w) <ρ(S_n-3,1^w(w*2), (See Chapter 2 Fig.5), if w _n-1² 3. In the third chapter, we prove the second largest spectrum of Halin graph.