Trees with maximum sum of the two largest Laplacian eigenvalues

On the sum of the two largest Laplacian eigenvalues of trees

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.

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...

Laplacian eigenvalues and the maximum cut problem

We introduce and study an eigenvalue upper bound p(G) on the maximum cut mc (G) of a weighted graph. The function ~o(G) has several interesting properties that resemble the behaviour of mc (G). The following results are presented. We show that q~ is subadditive with respect to amalgam, and additive with respect to disjoint sum and 1-sum. We prove that ~(G) is never worse that 1.131 mc(G) for a ...

Some remarks on the sum of the inverse values of the normalized signless Laplacian eigenvalues of graphs

Let G=(V,E), $V={v_1,v_2,ldots,v_n}$, be a simple connected graph with $%n$ vertices, $m$ edges and a sequence of vertex degrees $d_1geqd_2geqcdotsgeq d_n>0$, $d_i=d(v_i)$. Let ${A}=(a_{ij})_{ntimes n}$ and ${%D}=mathrm{diag }(d_1,d_2,ldots , d_n)$ be the adjacency and the diagonaldegree matrix of $G$, respectively. Denote by ${mathcal{L}^+}(G)={D}^{-1/2}(D+A) {D}^{-1/2}$ the normalized signles...