The Laplacian spectral radii of trees with degree sequences

Abstract

In this paper, we characterize all extremal trees with the largest Laplacian spectral radius in the set of all trees with a given degree sequence. Consequently, we also obtain all extremal trees with the largest Laplacian spectral radius in the sets of all trees of order n with the largest degree, the leaves number and the matching number, respectively.

Largest eigenvalues of the discrete p-Laplacian of trees with degree sequences

“…Furthermore, we show that the largest maximum eigenvalue in such classes of trees is strictly monotone with respect to some partial ordering of degree sequences. Thus, we extend a result that has been independently shown by Zhang [11] and Bıyıkoglu et al [4]. It is remarkable that this result is independent from the value of p. Although there is still little known about the p-Laplacian our result shows that it shares at least some of the properties with the combinatorial Laplacian.…”

The (signless) Laplacian spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices

“…For terminology and notation not defined here, we refer the readers to [1], [2], [4]- [6], [10], [12], [13], [17], [18] and the references therein.…”

The first Dirichlet eigenvalue of bicyclic graphs

“…, d n−1 ) is called graphic degree sequence if there exists a simple connected graph having π as its vertex degree sequence. Zhang [6] determined the extremal graphs which have largest Laplacian eigenvalues among all trees with a given graphic tree degree. Recently, the Dirichlet eigenvalues of graphs with boundary have received much attention (see [1], [3]- [5]), since the graph Laplacian can be regarded the discrete analog of the continuous Laplace operator on manifolds.…”