Improved bounds for the largest eigenvalue of trees Original Research Article

Let image be a tree with vertex set V. Let dv denotes the degree of v set membership, variant V. Let Δ = max{dv : v set membership, variant V}. Let u set membership, variant V such that du = Δ. Let k = eu + 1 where eu is the excentricity of u. For j = 1, 2, …, k − 2, letδj=max{dv:dist(v,u)=j}.We prove thatimageandimagewhere image and image are the largest eigenvalue of the Laplacian matrix and adjacency matrix of T, respectively. These bounds give better results than those obtained in [D. Stevanović, Linear Algebra Appl. 360 (2003) 35–42] except if δ1 = Δ.