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For a simple graph G on n vertices the Seidel Estrada index of G, denoted by $ SEE(G) $ SEE (G) , is defined as $ SEE(G)=\sum _{i=1}^ne^{\theta _i} $ SEE (G) = ∑ i = 1 n e θ i , where $ \theta _1,\ldots,\theta _n $ θ 1 , ... , θ n are the Seidel eigenvalues (the eigenvalues of the Seidel matrix) of G. In this paper, we find the maximum and minimum values of the Seidel Estrada index among all graphs with the fixed number of vertices. Our results confirm some conjectures on Seidel Estrada index of graphs that have been posed in [M. Hakimi-Nezhaad, M. Ghorbani, On the Estrada Index of Seidel Matrix, Mathematics Interdisciplinary Research5 (2020) 43–54]. [ABSTRACT FROM AUTHOR]

In this paper, we study that the algebraic multiplicity of the zero Laplacian eigenvalue of a connected uniform hypergraph. We give the algebraic multiplicity of the zero Laplacian eigenvalue of a hyperstar. For a loose hyperpath, we characterize the algebraic multiplicity of the zero Laplacian eigenvalue by the multiplicities of points in the affine variety defined by the Laplacian eigenvalue equations. We compute the algebraic multiplicities of the zero Laplacian eigenvalue of a loose hyperpath. We also show that the algebraic multiplicity of the zero Laplacian eigenvalue is not smaller than the number of irreducible components of the eigenvariety associated with the zero Laplacian eigenvalue for a loose hyperpath. [ABSTRACT FROM AUTHOR]

The dangling nodes are the nodes with no out-links in the web graph. It saves many computational costs if the dangling nodes are lumped into one node. In this paper, motivated by so many dangling nodes in web graphs, we mainly develop theoretical results for the modified hyperlink-induced topic search (MHITS) model by the lumping method, and the results are also suitable for the HITS model. There are three main findings in the theoretical analysis. First, MHITS can be lumped, although the matrix involved is not stochastic. Second, the hub vector of the nondangling nodes can be computed separately from dangling nodes, and the hub vector of the dangling nodes is parallel to a vector of all ones in MHITS. Third, the authority vector of the nondangling nodes is difficult to compute separately from dangling nodes. The numerical results not only show the feasibility and effectiveness of theoretical analyses, but also demonstrate that the lumped MHITS method can produce a better initial authority vector after the iteration of computing the hub vector. [ABSTRACT FROM AUTHOR]

An n-by-n real symmetric matrix $ H=[h_{i,j}] $ H = [ h i , j ] is said to be a Hankel matrix if $ h_{i,j}=h_{i-1,j+1} $ h i , j = h i − 1 , j + 1 , for each $ i=2,\ldots,n $ i = 2 , ... , n and $ j=1,\ldots,n-1 $ j = 1 , ... , n − 1. The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when a list $ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $ Λ = { λ 1 , ... , λ n } of real numbers is the spectrum of an n-by-n symmetric nonnegative matrix H. In this paper, we search for conditions on the list $ \Lambda =\{\lambda _{1},\ldots,\lambda _{n}\} $ Λ = { λ 1 , ... , λ n } for the matrix H to be Hankel. For n = 3, sufficient conditions are established. In particular, a necessary and sufficient condition is obtained if Λ is a list of three nonnegative numbers. Also, if $ \sum _{i=1}^{n}\lambda _{i}=0 $ ∑ i = 1 n λ i = 0 , we give conditions for realizability by a Hankel matrix. Finally, we present a special type of list that can serve as the spectrum of a Hankel nonnegative matrix with positive trace. Several of our results are constructive and provide a Hankel realizing matrix. [ABSTRACT FROM AUTHOR]

In this paper, we consider the spectral radius of signless p-Laplacian of a graph, which is a generalization of the quadratic form of the signless Laplacian matrix for p = 2. Let G n , β be the set of simple graphs of order n with a given matching number β. In this paper, the graphs maximizing the largest signless p-Laplacian eigenvalue among G n , β are obtained. [ABSTRACT FROM AUTHOR]

An n × n matrix C is said to be centrosymmetric if it satisfies the relation JCJ = C, where J is the n × n counteridentity matrix. Centrosymmetric matrices have a rich eigenstructure that has been studied extensively in the literature. Many results for centrosymmetric matrices have been generalized to wider classes of matrices that arise in a wide variety of disciplines. In this paper, we obtain interesting spectral properties for nonnegative centrosymmetric matrices. We show how to change one single eigenvalue, two or three eigenvalues of an n × n nonnegative centrosymmetric matrix without changing any of the remaining eigenvalues, the nonnegativity, or the centrosymmetric structure. Moreover, our results allow partially answer some known questions given by [Guo W. Eigenvalues of nonnegative matrices. Linear Algebra Appl. 266;1997:261–270] and by [Guo S, Guo W. Perturbing non-real eigenvalues of non-negative real matrices. Linear Algebra Appl. 426;2007:199–203]. Our proofs generate algorithmic procedures that allow one to compute a solution matrix. [ABSTRACT FROM AUTHOR]

In this paper, we consider the perturbation of a Hermitian matrix pair (H , M) , where H and M are non-singular and positive definite Hermitian matrices, respectively. A novel upper bound on a tangent of the angles between the eigenspaces of perturbed and unperturbed pairs is derived under a perturbation to the off-diagonal blocks of H. The rotation of the eigenspaces under a perturbation is measured in the matrix-dependent scalar product. We show that a tan ⁡ Θ bound for the standard eigenvalue problem is a special case of our new bound and that the obtained bound can be much sharper than the existing sin ⁡ Θ bounds. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem on the existence of perfect state transfer (PST for short) on semi-Cayley graphs over abelian groups (which are not necessarily regular), i.e. on the graphs having semiregular and abelian subgroups of automorphisms with two orbits of equal size. We stablish a characterization of semi-Cayley graphs over abelian groups having PST. As a result, we give a characterization of Cayley graphs over groups with an abelian subgroup of index 2 having PST, which improves the earlier results on Cayley graphs over abelian groups, dihedral groups and dicyclic group and determines Cayley graphs over generalized dihedral groups and generalized dicyclic groups having PST. [ABSTRACT FROM AUTHOR]

A graph G is k-edge-Hamiltonian if any collection of vertex-disjoint paths with at most k edges altogether belong to a Hamiltonian cycle in G. A graph G is k-Hamiltonian if for all S ⊆ V (G) with | S | ≤ k , the subgraph induced by V (G) ∖ S has a Hamiltonian cycle. These two concepts are classical extensions of the usual Hamiltonian graphs. In this paper, we present some spectral sufficient conditions for a graph to be k-edge-Hamiltonian and k-Hamiltonian in terms of the adjacency spectral radius as well as the signless Laplacian spectral radius. Our results could be viewed as slight extensions of the recent theorems proved by Li and Ning [Linear Multilinear Algebra. 2016;64:2252–2269], Nikiforov [Czechoslovak Math J. 2016;66:925–940] and Li et al. [Linear Multilinear Algebra. 2018;66:2011–2023]. Moreover, we shall prove a stability result for graphs being k-Hamiltonian, which could be regarded as a complement of two recent results of Füredi et al. [Discrete Math. 2017;340:2688–2690] and [Discrete Math. 2019;342:1919–1923]. [ABSTRACT FROM AUTHOR]

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic singular value and spectral distributions of matrices A n arising from virtually any kind of numerical discretization of differential equations (DEs). Indeed, when the mesh fineness parameter n tends to infinity, these matrices A n give rise to a sequence { A n } n , which often turns out to be a GLT sequence. In this paper, we provide an extension of the theory of GLT sequences: we show that any sequence of diagonal sampling matrices constructed from asymptotically uniform samples of an almost everywhere continuous function falls in the class of GLT sequences. We also detail a few representative applications of this result in the context of finite difference discretizations of DEs with discontinuous coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of computing the nearest linearly structured polynomial matrix with prescribed distinct eigenvalues. The problem deals with the computation of the minimal structured perturbation to the coefficient matrices so that the perturbed polynomial matrix has the prescribed eigenvalues and is the nearest to the given polynomial matrix. In recent years, a series of papers have been published on the perturbation of polynomial matrices with prescribed eigenvalues; however, a linear structure-preserving result does not exist so far to the best of our knowledge. In this paper, we have proposed an optimization based approach where we have reformulated the problem as a constrained optimization problem. Towards the end, a few numerical case studies are presented which demonstrate the efficiency and usefulness of our proposed method. [ABSTRACT FROM AUTHOR]

Let G be a graph. The energy of G, E (G) , is defined as the sum of absolute values of its eigenvalues. Here, it is shown that if G is a graph and { H 1 , ... , H k } is an edge partition of G, such that H 1 , ... , H k are spanning; then E (G) = ∑ i = 1 k E (H i) if and only if A i A j = 0 , for every 1 ⩽ i , j ⩽ k and i ≠ j , where A i is the adjacency matrix of H i . It was proved that if G is a graph and H 1 , ... , H k are subgraphs of G which partition edges of G, then E (G) ⩽ ∑ i = 1 k E (H i). In this paper we show that if G is connected, then the equality is strict, that is E (G) < ∑ i = 1 k E (H i). [ABSTRACT FROM AUTHOR]

A c-cyclic graph is a connected graph with n vertices and n + c−1 edges. In this paper, we consider the problem that: in the class of graphs, each of which is the complement graph of a c-cyclic graph with n vertices, which graph has the largest A α -spectral radius. We shall show that, for 0 ≤ c ≤ n − 4 , the extremal graph must be a graph with an isolated vertex. This implies that the maximum degree of G is larger, then the A α -spectral radius of its complement graph is also larger. We also determine the structure of the extremal graph with largest A α -spectral radius among the class of graphs, each of which is the complement graph of a c-cyclic graph with n vertices and maximum degree n−1 for 0 ≤ c ≤ n − 3 and n−2 for 0 ≤ c ≤ n − 4 , respectively. [ABSTRACT FROM AUTHOR]

Let G and H be equivalent cographs with reductions R G and R H , and suppose the vertices of R G and R H are labelled by the twin numbers t i of the k twin classes they represent. In this paper, we prove that G and H have at least k + ∑ i ∈ I (t i − 1) Laplacian eigenvalues in common, where I ⊆ { 1 , 2 , ... , k } is the indices of the twin classes whose types are identical in G and H. This confirms the conjecture proposed by Abrishami [A combinatorial analysis of the eigenvalues of the Laplacian matrices of cographs [Master's thesis]. Johns Hopkins University; 2019. Available from: ]. We also show that no two nonisomorphic equivalent cographs are cospectral with relation to the Laplacian matrix. [ABSTRACT FROM AUTHOR]

The universal adjacency matrix U of a graph Γ, with adjacency matrix A, is a linear combination of A, the diagonal matrix D of vertex degrees, the identity matrix I, and the all-1 matrix J with real coefficients, that is, U = c 1 A + c 2 D + c 3 I + c 4 J , with c i ∈ R and c 1 ≠ 0. Thus, in particular cases, U may be the adjacency matrix, the Laplacian, the signless Laplacian, and the Seidel matrix. In this paper, we develop a method for determining the universal spectra and bases of all the corresponding eigenspaces of arbitrary lifts of graphs (regular or not). As an example, the method is applied to give an efficient algorithm to determine the characteristic polynomial of the Laplacian matrix of the symmetric squares of odd cycles, together with closed formulas for some of their eigenvalues. [ABSTRACT FROM AUTHOR]

Let H be a connected graph with diameter μ and with distance matrix D. The reverse Wiener matrix of H is considered as μ (J − I) − D , where I and J are the identity and all-ones matrix, respectively. The reverse Wiener spectral radius of H is the greatest eigenvalue of its reverse Wiener matrix. In this paper we obtain some relations on reverse Wiener spectral radius of graphs. We show that among all trees the paths have maximum and the stars have minimum reverse Wiener spectral radius. [ABSTRACT FROM AUTHOR]

The formula for the spectral moment of graphs which is represented by the number of subgraphs is called the RNS spectral moment formula of graphs. In this paper, the research on the RNS spectral moment formula of graphs is generalized to hypergraphs. We give a formula for the spectral moment of a hypertree in terms of the number of some subhypertrees. Using this formula, we obtain the first 3k codegree coefficients of the characteristic polynomial of a k-uniform hypertree in terms of the number of some subhypertrees. [ABSTRACT FROM AUTHOR]

The energy of a graph is defined as the sum of the absolute values of all eigenvalues with respect to its adjacency matrix. Denote by Gn,m the set of all connected graphs having n vertices and m edges. Caporossi et al. [Caporossi G, Cvetkovi D, Gutman I, et al. Variable neighbourhood search for extremal graphs. 2. Finding graphs with external energy. J Chem Inf Comput Sci. 1999;39:984–996] conjectured that among all graphs in Gn,m, n ≥ 6 and n − 1 ≤ m ≤ 2(n − 2), the graphs with minimum energy are the star Sn with m−n + 1 additional edges all connected to the same vertices for m ≤ n + ⌊ (n − 7) / 2 ⌋ , and the bipartite graph with two vertices on one side, one of which is connected to all vertices on the other side, otherwise. In this paper, we provide a new approach to investigate the conjecture above. Especially, we determine the unique tetracyclic graph having minimum energy. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a class of tensor equations. Some explicit solutions to these equations are obtained. They are extensions of some results given by Lancaster. At the end of the article, these results are also employed to prove Lyapunov and Stein stability theorems for tensors. [ABSTRACT FROM AUTHOR]

Let Γ be a graph with adjacency matrix A. The transition matrix of Γ corresponding to A is defined by H (t) := exp ⁡ (− ι t A) , where ι = − 1 and t ∈ R . The graph is said to exhibit pretty good state transfer between a pair of vertices u and v if there exists a sequence of real numbers { t k } and a complex number γ with unit norm such that lim k → ∞ H (t k) e u = γ e v. In this paper, we explore pretty good state transfer on Cayley graphs over semi-dihedral groups by using the representations of such groups. We show that graphs Cay (S D 8 n , S) have pretty good state transfer for some suitable subsets S if n is a power of 2. Moreover, we present a sufficient and necessary condition for a non-integral graph Cay (S D 8 n , S) to admit pretty good state transfer. Some concrete constructions of Cayley graphs over semi-dihedral groups having PGST are also presented. [ABSTRACT FROM AUTHOR]

In Markov chain theory, stochastic matrices are used to describe inter-state transitions. Powers of such transition matrices are computed to determine the behaviour within a Markov system. For this, diagonalizable matrices are preferred because of their useful properties. The non-diagonalizable matrices are therefore undesirable. The aim is to determine a nearby diagonalizable matrix A ~ , starting from a non-diagonalizable matrix A. Previous studies tackled this problem, limited to 3 × 3 stochastic matrices. In this paper, these results are generalized for n × n stochastic matrices. Spectral properties of A are preserved in this process, such that A and A ~ have coinciding semisimple eigenvalues and coinciding corresponding eigenvectors. This problem is examined and solved in this study and an algorithm is presented to find such a diagonalizable matrix A ~ . [ABSTRACT FROM AUTHOR]

In this paper, we propose generalized inverse power methods with variable shifts for finding the smallest H-/Z-eigenvalue and associated H-/Z-eigenvector of symmetric tensors. The methods are guaranteed to always converge to a H-/Z-eigenpair. Furthermore, for an even order nonsingular symmetric M -tensor, the proposed method with any positive initial point always converges to the smallest H-eigenvalue. Numerical results are reported to illustrate that the proposed methods often can find the smallest H-/Z-eigenvalue instead of other H-/Z-eigenvalues of symmetric tensors. Moreover, we can always get the smallest H-eigenvalue for an even order nonsingular symmetric M -tensor. [ABSTRACT FROM AUTHOR]

A hypergraph is said to be oriented if each edge-vertex incidence has a label of +1 or −1. An oriented hypergraph is called incidence balanced if there exists a bipartition of the vertex set such that every edge intersects one part of the bipartition in positively incident vertices with the edge and the other part in negatively incident vertices with the edge. In this paper, we investigate the incidence balance of oriented hypergraphs and induced signed hypergraphs by means of the eigenvalues of associated matrices and tensors and provide a spectral method to characterize the incidence balanced oriented hypergraphs and their induced signed hypergraphs. [ABSTRACT FROM AUTHOR]

Let D be a simple digraph with eigenvalues z1, z2, ..., zn. The energy of D is defined as E (D) = ∑ i = 1 n | R e (z i) | , where Re(zi) is the real part of the eigenvalue zi. In this paper, a lower bound for the spectral radius of D will be established based on the number of subgraphs P 3 ↔ in D, improving some of the lower bounds that appear in the literature. Furthermore, this result allows us to obtain an upper bound for the energy of D. [ABSTRACT FROM AUTHOR]

In the last decade, several scholars proposed an unifying approach to study the spectral theories of the adjacency, Laplacian and signless Laplacian of graphs. The most general graph matrix is the universal adjacency matrixU = αA + βD + γJ + δI, where A, D, J, and I are the adjacency matrix of G, the degree matrix of G, the all-ones matrix, the identity matrix, respectively. Here, we consider M β = A + β D , with β ≥ 0, and we study the graphs belonging to some given class Γ maximizing the corresponding spectral radius ρ β . In particular, we consider connected graphs with prescribed c-cyclic degree sequence, c ∈ {0, 1, 2}, and the multicone graphs defined over them, where the multicone graph is the join of a clique with a given graph. The aim of this paper is to provide the best possible generalization of results to the spectral radius of M β (and the graph matrices related to it) of several well-known results for multicone graphs over connected graphs with prescribed c-cyclic degree sequence, where c ∈ {0, 1, 2}. [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]

Let S(G) be the Seidel matrix of a graph G. The Seidel energy of G, denoted by ES(G), is defined to be the sum of absolute values of all eigenvalues of the Seidel matrix S(G) of G. In this paper, we consider the change of Seidel energy of graphs due to edge deletion. It is proved that the Seidel energy of the tripartite Turán graph T(n, 3) of order n ≥ 9 is always increased when an edge is deleted. [ABSTRACT FROM AUTHOR]

For a simple graph G, the signless Laplacian matrix of G, denoted by Q(G), is defined as D(G) + A(G), where A(G) and D(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. By the smallest signless Laplacian eigenvalue of G, denoted by q′(G), we mean the smallest eigenvalue of Q(G). Let T(n, t) be the Turán graph on n vertices and t parts. In De Lima et al. (The clique number and the smallest Q-eigenvalue of graphs. Discrete Math. 2016;339:1744–1752), the authors posed the following conjecture:Conjecture. Let t ≥ 3 and let n be sufficiently large. If G is a Kt+1-free graph of order n and G ≠ T(n, t), then q′(G) < q′(T(n, t)).In this paper, first we disprove the above conjecture and then pose a new conjecture. [ABSTRACT FROM AUTHOR]

We study the transients of matrices in max-plus algebra. Our approach is based on the weak CSR expansion. Using this expansion, the transient can be expressed by max{T1, T2}, where T1 is the weak CSR threshold and T2 is the time after which the purely pseudoperiodic CSR terms start to dominate in the expansion. Various bounds have been derived for T1 and T2, naturally leading to the question which matrices, if any, attain these bounds. In the present paper, we characterize the matrices attaining two particular bounds on T1, which are generalizations of the bounds of Wielandt and Dulmage–Mendelsohn on the indices of non-weighted digraphs. This also leads to a characterization of tightness for the same bounds on the transients of critical rows and columns. The characterizations themselves are generalizations of those for the non-weighted case. [ABSTRACT FROM AUTHOR]

In this paper, we study the properties of extreme points of slice-stochastic tensors, and present an analogue of the Birkhoff–von Neumann theorem for slice-stochastic tensors. Some equivalent characterizations for a slice-stochastic tensor to be an extreme point are presented. The number of permutation tensors is also discussed. [ABSTRACT FROM AUTHOR]

Let A ∈ C n × n be a normal matrix with spectrum { λ i } i = 1 n , and let A ~ = A + E ∈ C n × n be a perturbed matrix with spectrum { λ ~ i } i = 1 n . If A ~ is still normal, the celebrated Hoffman–Wielandt theorem states that there exists a permutation π of { 1 , ... , n } such that (∑ i = 1 n | λ ~ π (i) − λ i | 2 ) 1 / 2 ≤ ∥ E ∥ F , where ∥ ⋅ ∥ F denotes the Frobenius norm of a matrix. This theorem reveals the strong stability of the spectrum of a normal matrix. However, if A or A ~ is non-normal, the Hoffman–Wielandt theorem does not hold in general. In this paper, we present new upper bounds for (∑ i = 1 n | λ ~ π (i) − λ i | 2 ) 1 / 2 , provided that both A and A ~ are general matrices. Some of our estimates improve or generalize the existing ones. [ABSTRACT FROM AUTHOR]

Let G be a graph having a unique perfect matching M, A (G) be the adjacency matrix of G and W (G) be the collection of all positive weight functions defined on the edge set of G in which each weight function w assigns weight 1 to each matching edge and a positive weight to each non-matching edge. The weighted graph G w satisfies the (− S R) property if for each eigenvalue of G w , its anti-reciprocal is also an eigenvalue of G w with the same multiplicity. In this paper, a class of graphs with a unique perfect matching M for which the diagonal entries of the inverse of the adjacency matrix of each graph are all zero is investigated. Furthermore, it is shown that no noncorona graph in this class satisfies the (− S R) property even for a single weight function w ∈ W (G). [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]

Let G be a graph and S (G) be the Seidel matrix of G. The Seidel energy E s (G) is defined to be the sum of the absolute value of all eigenvalues of the Seidel matrix of G. In this paper, we study some properties of the Seidel eigenvalues of G. Moreover, we present some lower and upper bounds for the Seidel energy of G. [ABSTRACT FROM AUTHOR]

Let G be an r-uniform hypergraph of order n. For each p ≥ 1 , the p-spectral radius λ (p) (G) is defined as λ (p) (G) := max | x 1 | p + ⋯ + | x n | p = 1 r ∑ { i 1 , ... , i r } ∈ E (G) x i 1 ⋯ x i r . The p-spectral radius was introduced by Keevash-Lenz-Mubayi, and subsequently studied by Nikiforov in 2014. The most extensively studied case is when p = r, and λ (r) (G) is called the spectral radius of G. The α-normal labelling method, which was introduced by Lu and Man in 2014, is effective method for computing the spectral radii of uniform hypergraphs. It labels each corner of an edge by a positive number so that the sum of the corner labels at any vertex is 1 while the product of all corner labels at any edge is α. Since then, this method has been used by many researchers in studying λ (r) (G). In this paper, we extend Lu and Man's α-normal labelling method to the p-spectral radii of uniform hypergraphs for p ≠ r ; and find some applications. [ABSTRACT FROM AUTHOR]

In this paper, we study the probability distribution of eigenvalues of the n × n Euclidean random matrix whose (i , j) t h entry is of the form f (∥ x i − x j ∥ 2) − u n δ i j ∑ k f (∥ x i − x k ∥ 2) for some real function f. Random points x i 's are independently distributed in the l p ellipsoid or on its surface in R N including the unit sphere S N − 1 , the simplex, the ordinary ellipsoid and the hyper-cube. Here u n is allowed to be any real number which includes the two most interesting cases u n = 0 and u n = 1. The limits of the empirical distribution of its eigenvalues are derived in two high dimensional settings: n / N → γ ∈ (0 , ∞) and n / N → 0 as both n and N go to infinity. By taking f (x) to be suitable functions, we also give the explicit limiting spectral distributions for some distance matrices whose entries are based on the Euclidean distance and the geodesic distance. [ABSTRACT FROM AUTHOR]

In row designs, n experimental units are arranged in time or along a line. Every experimental unit is allocated to one out of ν treatments. The purpose of this paper is to determine optimal row designs for estimating means, that is treatment effects, under the model of main effects with homogeneous population. The errors of the observations follow a first-order autoregressive process with parameter ρ. For ν = 3 , we show that the design d (3 m) : 1 3 2 ⋯ 1 3 2 ⏟ 3 m is D-optimal, when n = 3m, for any 0 < ρ < 1 and any m ≥ 2. Also, we find and construct the A-optimal designs, when n = 3m, for any 0 < ρ < 1 and any m ≥ 2. Moreover, we prove that the designs d (3 m + 1) : 1 3 2 ⋯ 1 3 2 ⏟ 3 m 1 and d (3 m + 2) : 1 2 3 ⋯ 1 2 3 ⏟ 3 m 12 are D-optimal, when n = 3m + 1 and n = 3m + 2, respectively, for any 0 < ρ < 1 and any m ≥ 1. Furthermore, we present the competing designs for some values of n, when ν ≥ 4 and − 1 < ρ < 1. [ABSTRACT FROM AUTHOR]

In this paper, we give refinements for the G-majorization triangle inequality (x + y) ↓ ≺ G x ↓ + y ↓ with vectors x , y of an Eaton triple and normal map (⋅) ↓ . For instance, we establish refinements for the majorization Ky Fan's eigenvalue inequality λ (x + y) ≺ m λ (x) + λ (y) with Hermitian matrices x and y and the eigenvalue operator λ (⋅). Likewise, we show refinements for the weak majorization Ky Fan's singular value inequality s (x + y) ≺ w s (x) + s (y) with complex matrices x and y, where s (⋅) stands for the singular values operator. In particular, we recover original Maligranda's refinement of the norm triangle inequality. [ABSTRACT FROM AUTHOR]

A Deza graph with parameters (n , k , b , a) is a k-regular graph with n vertices such that any two of its vertices have b or a common neighbours, where b ≥ a. In this paper we investigate spectra of Deza graphs. In particular, using the eigenvalues of a Deza graph we determine the eigenvalues of its children. Divisible design graphs are significant cases of Deza graphs. Sufficient conditions for Deza graphs to be divisible design graphs are given, a few families of divisible design graphs are presented and their properties are studied. Our special attention goes to the invertibility of the adjacency matrices of Deza graphs. [ABSTRACT FROM AUTHOR]

C-eigenvalues and C-eigenvectors of piezoelectric-type tensors play an important role in the piezoelectric effect and the converse piezoelectric effect. In this paper, with the help of components of general C-eigenvectors, we present a new class of inclusion sets to locate all general C-eigenvalues for a given general piezoelectric-type tensor. Finally, numerical examples are reported to be tighter than some existing results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a min-plus analogue of Jordan canonical forms of matrices. We first define the generalized eigenvector of a min-plus matrix A as an eigenvector of the kth power of A for some integer k. As in the conventional algebra, we consider a transformation matrix consisting of the basis of the generalized eigenspace. Then, we call a block diagonal matrix obtained from such transformation matrix a Jordan canonical form. In min-plus algebra, however, not all square matrices have Jordan canonical forms. We derive a necessary and sufficient condition of matrices having those in terms of graph theory. [ABSTRACT FROM AUTHOR]

In this paper, we almost completely determine the hypertrees with the second largest spectral radius among all hypertrees with m edges and given size of matching by using the matching polynomials and ordering of hypertrees. [ABSTRACT FROM AUTHOR]

In this paper, we obtain bounds for the extreme entries of the principal eigenvector of a hypergraph; these bounds are computed using the spectral radius and some classical parameters such as maximum and minimum degree. We also study inequalities involving the ratio and difference between the two extreme entries of this vector. [ABSTRACT FROM AUTHOR]

The distance spectral radius λ (G) of a strongly connected digraph G is the eigenvalue of its distance matrix D (G) with the largest modulus. Let G n d denote the set of strongly connected digraphs with order n and diameter d. In this paper, we completely determine the strongly connected digraphs minimizing λ (G) among all strongly connected digraphs with order n and diameter d, for d = 1, 2, 3, 4, 5, 6, 7, n−1. We also propose a conjecture about the minimum distance spectral radius among all strongly connected digraphs with given diameter d ≥ 8. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph with n vertices, m edges and having distance signless Laplacian eigenvalues ρ 1 ≥ ρ 2 ≥ ... ≥ ρ n ≥ 0. For 1 ≤ k ≤ n , let M k (G) = ∑ i = 1 k ρ i and N k (G) = ∑ i = 0 k − 1 ρ n − i be respectively the sum of k-largest distance signless Laplacian eigenvalues and the sum of k-smallest distance signless Laplacian eigenvalues of G. In this paper, we obtain the bounds for M k (G) and N k (G) in terms of the number of vertices n and the transmission σ (G) of the graph G. We propose a Brouwer-type conjecture for M k (G) and show that it holds for graphs of diameter one and graphs of diameter two for all k. As a consequence, we observe that the conjecture holds for threshold graphs and split graphs (of diameter two). We also show that it holds for k = n−1 and n for all graphs and for some k for r-transmission regular graphs. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some lower bounds on the smallest eigenvalue q (A ⋆ B) for the Fan product. The obtained bounds improve some existing results and generalize the corresponding results. Furthermore, some results of the comparison are also considered. [ABSTRACT FROM AUTHOR]

Let U = ( U 1 U 2 U 3 U 4 ) be a complex matrix of order n. In this paper, once a square submatrix U 1 is fixed, we present several necessary conditions on U 1 , U 2 , U 3 and U 4 , where U is a unitary matrix. Particularly, given a unitary matrix U = ( U 1 U 2 U 3 U 4 ) , we characterize the completions ( U 1 X Y U 4 ) and ( U 1 X Y Z ) to some unitary matrix, whenever U 1 and U 4 are partial isometries. [ABSTRACT FROM AUTHOR]