The Kth largest eigenvalue of a tree

Let λ_k(T) be the kth eigenvalue of a tree, [x] the largest integer not greater than x. It is shown that a tree with n vertices has λ_k(T)⩽√[(n-2)/k] for 2⩽k⩽[n/2], and this upper bound is best possible for n≡1 mod k.     

On conjectures involving second largest signless Laplacian eigenvalue of graphs

Let G=(V,E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix of G is L(G)=D(G)-A(G) and the signless Laplacian matrix of G is Q(G)=D(G)+A(G). In this paper we obtain a lower bound on the second largest signless Laplacian eigenvalue and an upper bound on the smallest signless Laplacian eigenvalue of G. In [5], Cvetkovi? et al. have given a series of 30 conjectures on Laplacian eigenvalues and signless Laplacian eigenvalues of G (see also [1]). Here we prove five conjectures.     

Some graphs whose second largest eigenvalue does not exceed 2

We determine all trees whose second largest eigenvalue does not exceed $\sqrt{2}$. Next, we consider two classes of bipartite graphs, regular and semiregular, with small number of distinct eigenvalues. For all graphs considered we determine those whose second largest eigenvalue is equal to $\sqrt{2}$. Some additional results are also given.     

An inequality involving the second largest and smallest eigenvalue of a distance-regular graph

For a distance-regular graph with second largest eigenvalue (resp., smallest eigenvalue) θ₁ (resp., θ_D) we show that (θ₁+1)(θ_D+1)?-b₁ holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.     

The largest eigenvalue of unicyclic graphs

Let G be a simple graph. Let λ₁(G) and μ₁(G) denote the largest eigenvalue of the adjacency matrix and the Laplacian matrix of G, respectively. Let Δ denote the largest vertex degree. If G has just one cycle, then

     

The spectra of some trees and bounds for the largest eigenvalue of any tree

Let T be an unweighted tree of k levels such that in each level the vertices have equal degree. Let n_k−j+1 and d_k−j+1 be the number of vertices and the degree of them in the level j. We find the eigenvalues of the adjacency matrix and Laplacian matrix of T for the case of two vertices in level 1 (n_k = 2), including results concerning to their multiplicity. They are the eigenvalues of leading principal submatrices of nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for these matrices are , 2 ? j ? k, while the diagonal entries are 0, …, 0, ±1, in the case of the adjacency matrix, and d₁, d₂, …, d_k−1, d_k ± 1, in the case of the Laplacian matrix. Finally, we use these results to find improved upper bounds for the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any given tree.     

A note on the second largest eigenvalue of the laplacian matrix of a graph

In this note, a lower bound for the second largest eigenvalue of the Laplacian matrix of a graph is given in terms of the second largest degree of the graph.     

On tricyclic graphs whose second largest eigenvalue does not exceed 1

A tricyclic graph of order n is a connected graph with n vertices and n + 2 edges. In this paper, all tricyclic graphs whose second largest eigenvalue does not exceed 1 are identified.     

The largest eigenvalue of a graph: A survey

This article is a survey of results concerning the largest eigenvalue (or index) of a grapn, catcgoiizeu as follows (1) inequalities lor the index, (2) graph with bounded index, (3) ordering graphs by their indices, (4) graph operations and modifications, (5) random graphs, (6) applications.     

Spectral characterization of graphs whose second largest eigenvalue is less than 1

Graphs with second largest eigenvalue λ₂?1 are extensively studied, however, whether they are determined by their adjacency spectra or not is less considered. In this paper we completely characterize all the connected bipartite graphs with λ₂<1 that are determined by their adjacency spectra. In addition, we prove that all the connected non-bipartite graphs with girth no less than 4 and λ₂<1 are determined by their adjacency spectra.     

The largest eigenvalue distribution of the laguerre unitary ensemble

We study the probability that all eigenvalues of the Laguerre unitary ensemble of n by n matrices are in(0, t), that is, the largest eigenvalue distribution. Associated with this probability, in the ladder operator approach for orthogonal polynomials, there are recurrence coefficients, namely, α_n(t) and β_n(t), as well as three auxiliary quantities, denoted by r_n(t), R_n(t), and σ_n(t). We establish the second order differential equations for both β_n(t) and r_n(t). By investigating the soft edge scaling limit when α = O(n) as n →∞ or α is finite, we derive a P_Ⅱ, the σ-form, and the asymptotic solution of the probability. In addition, we develop differential equations for orthogonal polynomials P_n(z) corresponding to the largest eigenvalue distribution of LUE and GUE with n finite or large. For large n,asymptotic formulas are given near the singular points of the ODE. Moreover, we are able to deduce a particular case of Chazy's equation for ρ(t) = Ξ′(t) with Ξ(t) satisfying the σ-form of P_Ⅳ or P_Ⅴ.     

On the third largest Laplacian eigenvalue of a graph

In this article, a sharp lower bound for the third largest Laplacian eigenvalue of a graph is given in terms of the third largest degree of the graph.     

Proof of conjecture involving the second largest signless Laplacian eigenvalue and the index of graphs

Let G = (V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q(G) = D(G) + A(G). In [5], Cvetkovi? et al. have given the following conjecture involving the second largest signless Laplacian eigenvalue (q₂) and the index (λ₁) of graph G (see also Aouchiche and Hansen [1]):

     

Extremal hexagonal chains concerning largest eigenvalue

In this paper, we define a roll-attaching operation of a hexagonal chain, and prove Gutman’s conjecture affirmatively by using the operation. The idea of the proof is also applicable to the results concerning extremal hexagonal chains for the Hosoya index and Merrifield-Simmons index     

On graphs whose second largest eigenvalue equals 1 – the star complement technique

The star complement technique is a spectral tool recently developed for constructing some bigger graphs from their smaller parts, called star complements. Here we first identify among trees and complete graphs those graphs which can be star complements for 1 as the second largest eigenvalue. Using the graphs just obtained, we next search for their maximal extensions, either by theoretical means, or by computer aided search.     

The largest eigenvalue of small rank perturbations of Hermitian random matrices

We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. We consider random Hermitian matrices with independent Gaussian entries M _ij ,i≤j with various expectations. We prove that the largest eigenvalue of such random matrices exhibits, in the large N limit, various limiting distributions depending on both the eigenvalues of the matrix and its rank. This rank is also allowed to increase with N in some restricted way. An erratum to this article is available at .     

On thek-th largest eigenvalue of the Laplacian matrix of a graph

In this paper, we give the upper bound and lower bound ofk-th largest eigenvalue λk of the Laplacian matrix of a graphG in terms of the edge number ofG and the number of spanning trees ofG. This research is supported by the National Natural Science Foundation of China (Grant No.19971086) and the Doctoral Program Foundation of State Education Department of China.     

An explicit formula for eigenvalues of Bethe trees and upper bounds on the largest eigenvalue of any tree

A Bethe tree B_d,k is a rooted unweighted of k levels in which the root vertex has degree equal to d, the vertices at level j(2?j?k-1) have degree equal to (d+1) and the vertices at level k are the pendant vertices. In this paper, we first derive an explicit formula for the eigenvalues of the adjacency matrix of B_d,k. Moreover, we give the corresponding multiplicities. Next, we derive an explicit formula for the simple nonzero eigenvalues, among them the largest eigenvalue, of the Laplacian matrix of B_d,k. Finally, we obtain upper bounds on the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any tree T. These upper bounds are given in terms of the largest vertex degree and the radius of T, and they are attained if and only if T is a Bethe tree.