Ordering trees with n vertices and matching number q by their largest Laplacian eigenvalues

Ordering trees with n vertices and matching number q by their largest Laplacian eigenvalues

Denote by Tn,q the set of trees with n vertices and matching number q. Guo [On the Laplacian spectral radius of a tree, Linear Algebra Appl. 368 (2003) 379–385] gave the tree in Tn,q with the greatest value of the largest Laplacian eigenvalue. In this paper, we give another proof of this result. Using our method, we can go further beyond Guo by giving the tree in Tn,q with the second largest va...

Ordering trees by their Laplacian spectral radii

Denote by Tn the set of trees on n vertices. Zhang and Li [X.D. Zang, J.S. Li, The two largest eigenvalues of Laplacian matrices of trees (in Chinese), J. China Univ. Sci. Technol. 28 (1998) 513–518] and Guo [J.M. Guo, On the Laplacian spectral radius of a tree, Linear Algebra Appl. 368 (2003) 379–385] give the first four trees in Tn, ordered according to their Laplacian spectral radii. In this...

ORDERING CACTI WITH n VERTICES AND k CYCLES BY THEIR LAPLACIAN SPECTRAL RADII

A graph is a cactus if any two of its cycles have at most one common vertex. In this paper, we determine the first sixteen largest Laplacian spectral radii together with the corresponding graphs among all connected cacti with n vertices and k cycles, where n > 2k + 8.

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

Trees that have greatest maximum p-Laplacian eigenvalue among all trees with a given degree sequence are characterized. It is shown that such extremal trees can be obtained by breadth-first search where the vertex degrees are non-increasing. These trees are uniquely determined up to isomorphism. Moreover, their structure does not depend on p.

On the sum of the two largest Laplacian eigenvalues of trees