On the spectral radius of trees with fixed diameter

Let T_{(n, d)} be the set of trees on n vertices with diameter d. In this paper, the first spectral radii of trees in the set T_{(n, d)} (3 ? d ? n − 4) are characterized.     

On weighted spectral radius of unraveled balls and normalized Laplacian eigenvalues

For a graph G, the unraveled ball of radius r centered at a vertex v is the ball of radius r centered at v in the universal cover of G. We obtain a lower bound on the weighted spectral radius of unraveled balls of fixed radius in a graph with positive weights on edges, which is used to present an upper bound on the $s^{\text{th}}$ (where $s\ge 2$) smallest normalized Laplacian eigenvalue of irregular graphs under minor assumptions. Moreover, when $s=2$, the result may be regarded as an Alon–Boppana type bound for a class of irregular graphs.     

The distance spectral radius of trees

The unique graphs with maximum distance spectral radius among trees with given number of vertices of maximum degree and among homeomorphically irreducible trees, respectively, are determined.     

On multiple eigenvalues of trees

Let T be a tree of order n>6 with μ as a positive eigenvalue of multiplicity k. Star complements are used to show that (i) if k>n/3 then μ=1, (ii) if μ=1 then, without restriction on has k+1 pendant edges that form an induced matching. The results are used to identify the trees with a non-zero eigenvalue of maximum possible multiplicity.     

On the Laplacian spectral radius of weighted trees with a positive weight set

The spectrum of weighted graphs is often used to solve the problems in the design of networks and electronic circuits. We first give some perturbational results on the (signless) Laplacian spectral radius of weighted graphs when some weights of edges are modified; we then determine the weighted tree with the largest Laplacian spectral radius in the set of all weighted trees with a fixed number of pendant vertices and a positive weight set. Furthermore, we also derive the weighted trees with the largest Laplacian spectral radius in the set of all weighted trees with a fixed positive weight set and independence number, matching number or total independence number.     

Semiregular trees with minimal Laplacian spectral radius

A semiregular tree is a tree where all non-pendant vertices have the same degree. Among all semiregular trees with fixed order and degree, a graph with minimal (adjacency/Laplacian) spectral radius is a caterpillar. Counter examples show that the result cannot be generalized to the class of trees with a given (non-constant) degree sequence.     

关于k树的谱半径

洪渊给出了谱半径最大的k树.该文进一步定义了关于k树的一个参数l(G),借之给出了谱半径达到第二大和第三大的k树.     

Matchings in starlike trees

Let m(G,k) be the number of k-matchings in the graph G. We write G₁ ⪯ G₂ if m(G₁, k) ≤ m(G₂, k) for all k = 1, 2,…. A tree is said to be starlike if it possesses exactly one vertex of degree greater than two. The relation T₁ ⪯ T₂ is shown to hold for various pairs of starlike trees T₁, T₂. The starlike trees (with a given number of vertices), extremal with respect to the relation ⪯, are characterized.     

On the distribution of Laplacian eigenvalues of trees

For a tree T$T$ with n$n$ vertices, we apply an algorithm due to Jacobs and Trevisan (2011) to study how the number of small Laplacian eigenvalues behaves when the tree is transformed by a transformation defined by Mohar (2007). This allows us to obtain a new bound for the number of eigenvalues that are smaller than 2. We also report our progress towards a conjecture on the number of eigenvalues that are smaller than the average degree.     

The Laplacian spectral radius of trees and maximum vertex degree

Let $\Delta \left(T\right)$ and $\mu \left(T\right)$ denote the maximum degree and the Laplacian spectral radius of a tree $T$, respectively. In this paper we prove that for two trees $T_1$ and $T_2$ on $n(n\ge 21)$ vertices, if $\Delta \left(T_1\right)>\Delta \left(T_2\right)$ and $\Delta \left(T_1\right)\ge ?\frac{11n}{30}?+1$, then $\mu \left(T_1\right)>\mu \left(T_2\right)$, and the bound “$\Delta \left(T_1\right)\ge ?\frac{11n}{30}?+1$” is the best possible. We also prove that for two trees $T_1$ and $T_2$ on $2k(k\ge 4)$ vertices with perfect matchings, if $\Delta \left(T_1\right)>\Delta \left(T_2\right)$ and $\Delta \left(T_1\right)\ge ?\frac{k}{2}?+2$, then $\mu \left(T_1\right)>\mu \left(T_2\right)$.     

On the joint spectral radius of matrices of order 2 with equal spectral radius

We provide explicit formulas for the joint spectral radius of certain classes of pairs of real matrices of order 2 with equal spectral radius.     

On the eigenvalues of a nonlinear spectral problem

We consider a nonlinear eigenvalue problem of the Sturm–Liouville type with conditions of the third kind, which describes the propagation of polarized electromagnetic waves in a plane dielectric waveguide. The equation is nonlinear in the unknown function, and the boundary conditions depend on the spectral parameter nonlinearly. We obtain an equation for the spectral parameter and formulas for the zeros of the eigenfunctions and show that the problem has at most finitely many isolated eigenvalues.     

On the distance spectral radius of cacti

A cactus is a connected graph in which any two cycles have at most one common vertex. In this article, we determine the unique graph with minimal distance spectral radius in the class of all cacti with n vertices and k cycles. Also, we determine the unique graph with minimal distance spectral radius in the class of all cacti with n vertices and r pendent vertices. Moreover, we determine the class of cacti in which the maximal distance spectral radius among all cacti with n vertices and k cycles is attained.     

On the spectral radius of functional operators

An estimate of the spectral radius of functional operators generated by operators of multiplication and shift operators in the space of continuous vector functions on the interval is given. It is assumed that shifts have fixed points only at both ends of the interval and belong to one type, i.e., they are either left or right shifts.