Search

We consider the spectral radius of a large random matrix X with independent, identically distributed entries. We show that its typical size is given by a precise three-term asymptotics with an optimal error term beyond the radius of the celebrated circular law. The coefficients in this asymptotics are universal but they differ from a similar asymptotics recently proved for the rightmost eigenvalue of X in Cipolloni et al., Ann. Probab. 51(6), 2192–2242 (2023). To access the more complicated spectral radius, we need to establish a new decorrelation mechanism for the low-lying singular values of X − z for different complex shift parameters z using the Dyson Brownian Motion. [ABSTRACT FROM AUTHOR]

The arithmetic-geometric spectral radius of a graph G is the largest eigenvalue of the arithmetic-geometric matrix of G whose (u, v)-entry is d u + d v 2 d u d v if u and v are adjacent and 0 otherwise for u , v ∈ V (G) , where d u denotes the degree of vertex u in G. We determine the graphs with the largest and the next largest arithmetic-geometric spectral radii over all n-vertex bicyclic graphs with n ≥ 5 . [ABSTRACT FROM AUTHOR]

We classify maximal digraphs whose Hermitian spectral radius is at most 2. [ABSTRACT FROM AUTHOR]

The ε-spectral radius of a connected graph is the largest eigenvalue of its eccentricity matrix. In this paper, we identify the unique n-vertex tree with diameter 4 and matching number 5 that minimizes the ε-spectral radius, and thus resolve a conjecture proposed in [W. Wei, S. Li, L. Zhang, Characterizing the extremal graphs with respect to the eccentricity spectral radius, and beyond, Discrete Math. 345 (2022) 112686]. [ABSTRACT FROM AUTHOR]

The distance Laplacian matrix of a connected graph G is defined as L (G) = T r (G) − D (G) , where Tr(G) is the diagonal matrix of the vertex transmissions in G and D(G) is the distance matrix of G. The largest eigenvalue of L (G) is called the distance Laplacian spectral radius of G. In this paper, we determine the graphs with the maximum distance Laplacian spectral radius and the minimum distance Laplacian spectral radius among all the bicyclic graphs with given order, respectively. [ABSTRACT FROM AUTHOR]

A pendant path of length r with r ≥ 1 in a graph G is a path u 0 u 1 ⋯ u r in G such that the degree of u 0 is one, the degree of u r is at least two, and if r ≥ 2 , then the degree of u i is two for any i with 1 ≤ i ≤ r - 1 . The distance spectral radius of a connected graph is the largest eigenvalue of the distance matrix of the graph. We determine the unique tree that maximizes (respectively minimizes) the distance spectral radius over all trees on n vertices with k pendant paths of length r, and also determine the unique graph that minimizes the distance spectral radius over all connected graphs on n vertices with k pendant paths of length r, where k , r ≥ 1 and k r < n - 1 . [ABSTRACT FROM AUTHOR]

Let D be a digraph of order n and let A (D) be the adjacency matrix of D. Let D e g (D) be the diagonal matrix of vertex out-degrees of D. For any real α ∈ [ 0 , 1 ] , the generalized adjacency matrix A α (D) of the digraph D is defined as A α (D) = α D e g (D) + (1 − α) A (D). The largest modulus of the eigenvalues of A α (D) is called the generalized adjacency spectral radius or the A α -spectral radius of D. In this paper, we obtain some lower bounds for the spectral radius of A α (D) in terms of the number of vertices, the number of arcs and the number of closed walks of the digraph D. The extremal graphs attained these lower bounds are determined. We also obtain some bounds for the spectral radius of A α (D) in terms of the vertex out-degrees, the vertex average 2-out-degrees of the vertices of D and parameter α. We characterize the extremal digraphs attaining these bounds. [ABSTRACT FROM AUTHOR]

Since the eigenvalue problem of real supersymmetric tensors was proposed, there have been many results regarding the numerical algorithms for computing the spectral radius of nonnegative tensors, among which the NQZ method is the most studied. However, the NQZ method is only suitable for calculating the spectral radius of a special weakly primitive tensor, or a weakly irreducible primitive tensor that satisfies certain conditions. In this paper, by means of diagonal similarrity transformation of tensors, we construct a numerical algorithm for computing the spectral radius of nonnegative tensors with the aid of power functions. This algorithm is suitable for the calculation of the spectral radius of all weakly irreducible nonnegative tensors. Furthermore, it is efficient and can be widely applied. [ABSTRACT FROM AUTHOR]

We prove that every nodal domain of an eigenfunction of the Laplacian of eigenvalue λ on a d -dimensional closed Riemannian manifold contains a ball of radius c λ − 1 2 (log ⁡ λ) − d − 2 2 . This ball is centered at a point at which the eigenfunction attains its maximum in absolute value within the nodal domain. [ABSTRACT FROM AUTHOR]

Let G be a digraph and A (G) be the adjacency matrix of G. Let D (G) be the diagonal matrix with outdegrees of vertices of G. For any real α ∈ [ 0 , 1 ] , define the matrix A α (G) as A α (G) = α D (G) + (1 − α) A (G). The largest modulus of the eigenvalues of A α (G) is called the A α spectral radius of G. In this paper, we determine the digraphs which attain the maximum (or minimum) A α spectral radius among all strongly connected digraphs with given parameters such as girth, clique number, vertex connectivity or arc connectivity. We also propose an open problem. [ABSTRACT FROM AUTHOR]

For any real number α ∈ [ 0 , 1 ] and a graph G , Nikiforov [20] defined matrix A α (G) as A α (G) = α D (G) + (1 − α) A (G). The A α - spectral radius of G is the largest eigenvalue of A α (G). In this paper, we determine the first three largest extremal graphs in Halin graphs by their A α - spectral radii when 1 2 ≤ α < 1. For 0 ≤ α ≤ 1 , the 3-regular Halin graph alone minimizes the A α - spectral radius for Halin graphs. [ABSTRACT FROM AUTHOR]

The work described in this paper is concerned with providing a rational asymptotic analysis for the partial wrinkling bifurcation of a thin elastic hemispherical segment in which the upper rim experiences in-plane circular shearing relative to the other circular edge. The mathematical structure of the associated complex-valued boundary eigenvalue problem is revealed by using the method of matched asymptotic expansions. Our key result is a three-term asymptotic formula for the critical load in terms of a suitable small parameter proportional to the ratio between the thickness and the radius of the shell. Comparisons of this formula with direct numerical simulations provide further insight into the range of validity of the results derived herein. [ABSTRACT FROM AUTHOR]

The ABC spectral radius of a graph G is the largest eigenvalue of the ABC matrix of G , whose (u , v) -entry is d u + d v − 2 d u d v if u v ∈ E (G) and 0 otherwise for u , v ∈ V (G) , where d u denotes the degree of u in G , and V (G) (E (G) , respectively) denotes the vertex (edge, respectively) set of G. The unicyclic graphs with the first four largest ABC spectral radii are determined. [ABSTRACT FROM AUTHOR]

Let G be a connected irregular graph on n vertices with maximum degree Δ and diameter D. The spectral radius of G, which is denoted by ρ (G) , is the largest eigenvalue of the adjacency matrix of G. In this paper, we study the lower bound of Δ - ρ (G) . As a result, a new lower bound is obtained which improves the known lower bounds of Δ - ρ (G) . [ABSTRACT FROM AUTHOR]

The ABC matrix of a graph G , proposed by Estrada in 2017, can be regarded as a weighed version of adjacency matrices of graphs, in which the (u , v) -entry is equal to d u + d v − 2 d u d v if uv is an edge of a graph G , and 0 otherwise, where d u represents the degree of u in G. The research about ABC spectral radius (largest eigenvalue of ABC matrix) of graphs is rather active in recent years. In this paper, we characterize the bicyclic graphs with the first two maximum ABC spectral radii, which are just the unique two bicyclic graphs of order n with maximum degree n − 1 if n ≥ 7. [ABSTRACT FROM AUTHOR]

The spectral radius ρ (G) of a graph G is the largest eigenvalue of its adjacency matrix A (G). In this paper, we first give a sharp lower bound of the spectral radius of a graph G in terms of the degree and the average 2-degree which generalizes a result of Das and Kumar (2003), and then as an application, a sharp upper bound of the energy of G is given which generalizes a result of Liu et al. (2007). [ABSTRACT FROM AUTHOR]

We consider linear time-invariant differential equations with the Hukuhara derivative. We prove that if a solution of such an equation has a nonempty interior at the initial time, then the Lyapunov exponents of the radii of the inscribed and circumscribed spheres of this solution are strict and are equal to the minimum and maximum absolute values, respectively, of the eigenvalues of the coefficient matrix of the system. [ABSTRACT FROM AUTHOR]

The distance Laplacian matrix of a connected graph G is defined as ℒ G = Tr G − D G , where D G is the distance matrix of G and Tr G is the diagonal matrix of vertex transmissions of G. The largest eigenvalue of ℒ G is called the distance Laplacian spectral radius of G. In this paper, we determine the graphs with maximum and minimum distance Laplacian spectral radius among all clique trees with n vertices and k cliques. Moreover, we obtain n vertices and k cliques. [ABSTRACT FROM AUTHOR]

We employ the concept of 2-degree of the vertices and, more generally, the number of walks between two vertices to introduce new upper and lower bounds for the spectral radius and the smallest eigenvalue of a graph. We, further, show how upper and lower bounds for the spectral radius are better than previous bounds in some cases. [ABSTRACT FROM AUTHOR]

For 0 ≤ α ≤ 1, Nikiforov proposed to study the spectral properties of the family of matrices Aα(G) = αD(G)+(1 - α)A(G) of a graph G, where D(G) is the degree diagonal matrix and A(G) is the adjacency matrix of G. The α-spectral radius of G is the largest eigenvalue of Aα(G). For a graph with two pendant paths at a vertex or at two adjacent vertices, we prove results concerning the behavior of the α-spectral radius under relocation of a pendant edge in a pendant path. We give upper bounds for the α-spectral radius for unicyclic graphs G with maximum degree ∆ ≥ 2, connected irregular graphs with given maximum degree and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with the second largest α-spectral radius among trees, and the unique tree with the largest α-spectral radius among trees with given diameter. We also determine the unique graphs so that the difference between the maximum degree and the α-spectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively. [ABSTRACT FROM AUTHOR]

In this paper we consider the spectral radii of a special set system called hypertrees, i.e. connected acyclic hypergraphs. By employing the theory of matching polynomial of hypertrees and ordering of hypertrees, we determine the largest spectral radius of hypertrees with m edges and given size of matching. [ABSTRACT FROM AUTHOR]

In this paper, we determine the unique graph among complements of unicyclic graphs which maximizes the distance spectral radius. We also characterize the unique graph among complements of unicyclic graphs of diameter three which maximizes the least distance eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some bounds for the ABC spectral radius of general graphs. In continuing, we get some properties of the ABC eigenvalues of graphs, and then we establish some new bounds for the ABC energy E A B C. Finally, we set up the correlation between the ABC energy and other classes of graph energies. [ABSTRACT FROM AUTHOR]

A connected graph G is spectrally non-redundant if the spectral radiuses of the connected induced subgraphs of G are all different. As observed by Fernandes, Júdice, and Trevisan (2017), for such graphs the number of connected induced subgraphs is equal to the number of complementarity eigenvalues. This note is an attempt at quantifying the so-called spectral redundancy phenomenon. Such a phenomenon occurs when there is a substantial amount of repetition among the spectral radiuses of the connected induced subgraphs. [ABSTRACT FROM AUTHOR]

We present progress on the problem of asymptotically describing the adjacency eigenvalues of random and complete uniform hypergraphs. There is a natural conjecture arising from analogy with random matrix theory that connects these spectra to that of the all-ones hypermatrix. Several of the ingredients along a possible path to this conjecture are established, and may be of independent interest in spectral hypergraph/hypermatrix theory. In particular, we provide a bound on the spectral radius of the symmetric Bernoulli hyperensemble, and show that the spectrum of the complete k -uniform hypergraph for k = 2 , 3 is close to that of an appropriately scaled all-ones hypermatrix. [ABSTRACT FROM AUTHOR]

The spectral radius of a uniform hypergraph G is the maximum modulus of the eigenvalues of the adjacency tensor of G. For k ≥ 2 , among connected k -uniform hypergraphs with m ≥ 1 edges, we show that the k -uniform loose path with m edges is the unique one with smallest spectral radius, and we also determine the unique ones with second smallest spectral radius when m ≥ 2. [ABSTRACT FROM AUTHOR]

Core results about the algebraic analysis of two-grid methods are extended in relations bounding the field of values (or numerical range) of the iteration matrix. On this basis, bounds are obtained on its norm and numerical radius, leading to rigorous convergence estimates. Numerical illustrations show that the theoretical results deliver qualitatively good predictions, allowing one to anticipate success or failure of the two-grid method. They also indicate that the field of values and the associated numerical radius are much more reliable convergence indicators than the eigenvalue distribution and the associated spectral radius. On this basis, some discussion is developed about the role of local Fourier or local mode analysis for nonsymmetric problems. [ABSTRACT FROM AUTHOR]

The resistance matrix R = R (G) of G is a matrix whose (i, j)th entry is equal to the resistance distance r G (v i , v j) . The resistance distance spectral radius, ∂ 1 R (G) , of G is the largest eigenvalues of R. In this paper, we obtain several edge-grafting transformations which are decreasing and/or increasing resistance distance spectral radius. As applications of these transformations, some lower and upper bounds on resistance distance spectral radius are determined and the corresponding extremal graphs are characterized. [ABSTRACT FROM AUTHOR]

Let D denote the smallest disc containing all eigenvalues of the matrix A. Without knowing the eigenvalues of A, we can estimate the spread of A and the radius of D. Some new bounds for the radius of D and the spread of A are given. These bounds involve the entries of A. Also sufficient conditions for equality are obtained for some inequalities. New proofs of some known results are presented, too. [ABSTRACT FROM AUTHOR]

This paper deals with various extremal problems involving two important parameters associated to a connected graph, say G. The first parameter is the largest complementarity eigenvalue: it is denoted by ϱ (G) and it is simply the spectral radius or index of the graph. Next in importance comes the second largest complementarity eigenvalue: it is denoted by ϱ 2 (G) and it is equal to the largest spectral radius among the children of G. By definition, a child or vertex-deleted connected subgraph of G is an induced subgraph obtained by removing a noncut vertex from G. In the first part of this work, we address the problem of identifying the eldest children and the youngest parents of G. We also analyze the uniqueness of such children and parents. An eldest child of G is a child whose spectral radius attains the value ϱ 2 (G) . The concept of youngest parent is somewhat dual to that of eldest child. The second part of this work is about minimization and maximization of the functions ϱ 2 and ϱ - ϱ 2 on special classes of connected graphs. We establish several new results and propose a number of conjectures. [ABSTRACT FROM AUTHOR]

This article has results of four types. We show that the first eigenvalue λ1(Ω) of the weighted Laplacian of a bounded domain with smooth boundary can be obtained by S. Sato's iteration scheme of the Green operator, taking the limit λ1(Ω) ࣪. Then, we study the L1(Ω, μ)-moment spectrum of Ω in terms of iterates of the Green operator G, extending the work of McDonald--Meyers to the weighted setting. As corollary, we obtain the first eigenvalue of a weighted bounded domain in terms of the L¹(Ω, μ)- moment spectrum, generalizing the work of Hurtado--Markvorsen--Palmer. Finally, we study the radial spectrum σrad(Bh(o, r)) of rotationally invariant geodesic balls Bh(o, r) of model manifolds. We prove an identity relating the radial eigenvalues of σrad(Bh(o, r)) to an isoperimetric quotient, i.e., Σ 1/λirad = ∫ V(s)/S9s)ds, V(s) = vol(Bh (o,s)) and S(s) = vol(∂Bh(o, s)). We then consider a proper minimal surface M ... ℝ³ and the extrinsic ball Ω = M ∩ BR3 (o, r). We obtain upper and lower estimates for the series Σλi-2 (Ω) in terms of the volume vol(Ω) and the radius r of the extrinsic ball Ω. [ABSTRACT FROM AUTHOR]

We obtain a sharp lower bound on the eigenvalues of the Laplace–Beltrami operator on a hyperbolic surface with injectivity radius bounded from below. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a simple connected graph of order n and size m , and let ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n − 1 > ρ n = 0 , be the normalized Laplacian eigenvalues of G. The greatest eigenvalue, ρ 1 , is referred to as the normalized Laplacian spectral radius. The Laplacian incidence energy, the Randić energy, and Kemeny's constant of G , are defined by L I E (G) = ∑ i = 1 n − 1 ρ i , R E (G) = ∑ i = 1 n | ρ i − 1 | , and K (G) = ∑ i = 1 n − 1 1 ρ i , respectively. New lower bounds for the normalized Laplacian spectral radius are obtained. The acquired bounds are then used to obtain new bounds for the L I E (G). In addition, some relations between L I E (G) , K (G) , R E (G) are determined. [ABSTRACT FROM AUTHOR]

A fractional matching of a graph G is a function f giving each edge a number in [ 0 , 1 ] so that ∑ e ∈ Γ (v) f (e) ≤ 1 for each v ∈ V (G) , where Γ (v) is the set of edges incident to v. The fractional matching number of G, written ν ∗ (G) , is the maximum of ∑ e ∈ E (G) f (e) over all fractional matchings. In this paper, we study the connections between the fractional matching number and the Laplacian spectral radius of a graph. We also give some sufficient spectral conditions for the existence of a fractional perfect matching. [ABSTRACT FROM AUTHOR]

For a real number α ∈ [ 0 , 1 ] , the A α -matrix of a graph G is defined to be A α (G) = α D (G) + (1 − α) A (G) , where A (G) and D (G) are the adjacency matrix and degree diagonal matrix of G , respectively. The A α -spectral radius of G , denoted by ρ α (G) , is the largest eigenvalue of A α (G). In this paper, we consider the upper bound of the A α -spectral radius ρ α (G) , also we give some upper bounds for the second largest eigenvalue of A α -matrix. [ABSTRACT FROM AUTHOR]

We show that for a positive proportion of Laplace eigenvalues λ j , the associated Hecke–Maass L -functions L (s , u j) approximate with arbitrary precision any target function f (s) on a closed disc with centre in 3/4 and radius r < 1 / 4. The main ingredients in the proof are the spectral large sieve of Deshouillers–Iwaniec and Sarnak's equidistribution theorem for Hecke eigenvalues. [ABSTRACT FROM AUTHOR]

Using the theory of equitable decompositions it is possible to decompose a matrix M appropriately associated with a given graph. The result is a collection of smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix M. This is done by decomposing the matrix over a graph symmetry. Previously it was shown that a matrix can be equitably decomposed over any uniform, basic, or separable automorphism. Here we extend this theory to show that it is possible to equitably decompose a matrix over any automorphism of a graph, without restriction. Moreover, we give a step-by-step procedure which can be used to generate such a decomposition. We also prove under mild conditions that if a matrix M is equitably decomposed the resulting divisor matrix, which is the divisor matrix of the associated equitable partition, will have the same spectral radius as the original matrix M. [ABSTRACT FROM AUTHOR]

Abstract M-eigenvalues of a fourth-order partially symmetric tensor play an important role in the nonlinear elastic material analysis and the entanglement problem in quantum physics. But it is not easy to get the M-eigenvalues exactly, especially for high dimensional fourth-order partially symmetric tensors. In this paper, we give two M-eigenvalue inclusion intervals for a fourth-order partially symmetric tensor, which provide two upper bounds for the M-spectral radius. As an application, taking these upper bounds as the parameter τ in WQZ-algorithm presented by Wang et al. (2009), respectively, can make the generated sequence by this algorithm more rapidly converge to a good approximation of the largest M-eigenvalue of a fourth-order partially symmetric tensor. [ABSTRACT FROM AUTHOR]

Abstract A ray pattern matrix is a matrix whose nonzero entries are all unimodular. An n × n ray pattern matrix A is said to require the weak Perron–Frobenius property if for any n × n entrywise positive matrix K , the spectral radius ρ (K ∘ A) is one of the eigenvalues of K ∘ A ; to require the strong Perron–Frobenius property if ρ (K ∘ A) is an algebraically simple eigenvalue of K ∘ A and the left and right eigenvectors associated to ρ (K ∘ A) are strictly nonzero. In this paper, necessary and sufficient conditions for ray pattern matrices requiring the weak/strong Perron–Frobenius property are given. [ABSTRACT FROM AUTHOR]

This paper presents a novel algorithm to compute the real stability radius for a linear delay system of retarded type with multiple delays. The real stability radius is the distance to instability, measured as the minimal real-valued perturbation that renders the system unstable. Our method is based on characterizing this distance to instability as the inverse of the global maximum of a real structured singular value function. We develop a criss-cross type algorithm that globally converges to this maximum, and whose convergence rate seems to be superlinear and sometimes quadratic in numerical experiments. The algorithm exploits that the intersections of these singular value functions with constant functions can be written as purely imaginary eigenvalues of certain delay eigenvalue problems (DEP) with positive and negative delays. This is an extension of the well-known linear case (without delays) where this results in algebraic eigenvalue problems. In addition, a novel numerical solver to compute all the imaginary eigenvalues of this DEP is also presented. It combines an approximation using spectral discretizations and an automatic procedure to determine the required number of discretization points. Finally, due to the presence of multiple eigenvalues at the maximum, these approximations are corrected with a block-Newton algorithm for nonlinear eigenvalue problems. [ABSTRACT FROM AUTHOR]

A new Brauer-type Z-eigenvalue inclusion set for tensors is given and proved to be tighter than those in Wang et al. (Discrete Contin. Dyn. Syst., Ser. B. 22(1), 187-198, 2017). Based on this set, a sharper upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Numerical examples are reported to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this work, a rigorous mathematical method to calculate the order parameter in a binary amorphous or liquid alloy, is established even when the alloy is made up of very differently sized atoms. This method, based on the study of the eigenstates of the exchange operator of the alloy components, allows revealing different spherical layers whose inner and outer radii are solutions of a certain equation. These layers have no constant thickness but contain numbers of atoms of each species that are exactly in the ratio of the concentrations of these two species. Thus, each layer has exactly the average concentration of the alloy, allowing to properly formalize the order parameter. Thus the order parameter is formalized in each of these layers. This method makes possible to retrieve the subtle Bhatia-Thornton formalism [Phys. Rev., B2, 3004 (1970)] used for the description of binary alloys. We discuss the link between the zero q value of concentration-concentration Bhatia-Thornton partial structure factor and the numbers of neighbours. We apply our method to study the order in the Bi 0.3 Ga 0.7 liquid alloy. We are able to determine eight layers and their order parameter on the interatomic distance-range [0, 15 Å]. Highlights • Rigorous mathematical method to calculate the order parameter in binary liquid alloy • Very accurate determination of the layers and their order parameter at any distance • The stoichiometry of the determined layers is always equal to the alloy ones • Method independent of size or chemical effects • Retrieving the Bhatia-Thornton formalism by a different physical approach [ABSTRACT FROM AUTHOR]

The distance spectral radius of a connected hypergraph is the largest eigenvalue of its distance matrix. We propose some graft transformations that decrease or increase the distance spectral radius of a connected hypergraph that is not necessarily uniform. Then we determine the unique hypertrees with minimum and maximum distance spectral radius, respectively, among hypertrees on n vertices with m edges, where . We also determine the unique hypertrees with the first three smallest (largest, respectively) distance spectral radii among hypertrees on vertices. [ABSTRACT FROM AUTHOR]

In this paper, we establish a tight sufficient condition for the Hamiltonicity of graphs with large minimum degree in terms of the signless Laplacian spectral radius and characterize all extremal graphs. Moreover, we prove a similar result for balanced bipartite graphs. Additionally, we construct infinitely many graphs to show that results proved in this paper give new strength for one to determine the Hamiltonicity of graphs. [ABSTRACT FROM AUTHOR]

Copyright of Basic Sciences Journal of Textile Universities / Fangzhi Gaoxiao Jichu Kexue Xuebao is the property of Basic Sciences Journal of Textile Universities and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let k≥1 and n1,…,nk≥1 be some integers. Let S(n1,…,nk) be a tree T such that T has a vertex v of degree k and T\v is the disjoint union of the paths Pn1,…,Pnk, that is T\v≅Pn1∪⋯∪Pnk so that every neighbor of v in T has degree one or two. The tree S(n1,…,nk) is called starlike tree, a tree with exactly one vertex of degree greater than two, if k≥3. In this paper we obtain the eigenvalues of starlike trees. We find some bounds for the largest eigenvalue (for the spectral radius) of starlike trees. In particular we prove that if k≥4 and n1,…,nk≥2, then k-1k-2<λ1(S(n1,…,nk)), where λ1(T) is the largest eigenvalue of T. Finally we characterize all starlike trees that all of whose eigenvalues are in the interval (-2,2). [ABSTRACT FROM AUTHOR]

In this paper, we give upper bounds on Z t -spectral radius of a tensor A ( t = 1 , 2 ) , which extend the upper bounds of Brauer to tensors. Moreover, an upper bound on the Z 1 -spectral radius is proposed via modulus sum of the entries of certain dimension of A , which improves the upper bound given by Li et al. Numerical experiments are given to illustrate the utility of the upper bound. [ABSTRACT FROM AUTHOR]

Consider a truncated circular unitary matrix which is a p n by p n submatrix of an n by n circular unitary matrix by deleting the last n − p n columns and rows. Jiang and Qi [11] proved that the maximum absolute value of the eigenvalues (known as spectral radius) of the truncated matrix, after properly normalized, converges in distribution to the Gumbel distribution if p n / n is bounded away from 0 and 1. In this paper we investigate the limiting distribution of the spectral radius under one of the following four conditions: (1). p n → ∞ and p n / n → 0 as n → ∞ ; (2). ( n − p n ) / n → 0 and ( n − p n ) / ( log ⁡ n ) 3 → ∞ as n → ∞ ; (3). n − p n → ∞ and ( n − p n ) / log ⁡ n → 0 as n → ∞ and (4). n − p n = k ≥ 1 is a fixed integer. We prove that the spectral radius converges in distribution to the Gumbel distribution under the first three conditions and to a reversed Weibull distribution under the fourth condition. [ABSTRACT FROM AUTHOR]

This work concerns with buckling and vibration analysis of composite plates based on a transverse shear theory. A numerical scheme is introduced to determine the angular frequencies and critical buckling loads of such plates. Moving least square differential quadrature method is employed to reduce the problem to that of eigen value problem. The accuracy and efficiency of the proposed scheme is examined with different computational characteristics, (radius of support domain, basis completeness order, and scaling factors). The obtained results agreed, at less execution time, with the previous ones. Further, a parametric study is introduced to investigate the influence of elastic and geometric characteristics, (Young's modulus gradation ratio, shear modulus gradation ratio, Poisson's ratio, loading parameter, and aspect ratio), of the composite on the values of critical buckling load, natural frequencies, and behavior of mode shape functions. [ABSTRACT FROM PUBLISHER]

Let G be a connected graph of order n and D ( G ) be its distance matrix. In this paper, we characterize the unique graphs whose distance spectral radius attains the maximum and minimum among all complements of trees. Furthermore, we determine the unique graphs whose least distance eigenvalues attains the maximum and minimum among all complements of trees. [ABSTRACT FROM AUTHOR]