Abstract A list Λ = { λ 1 , λ 2 , … , λ n } \Lambda =\left\{{\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{n}\right\} of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix A A . The list Λ \Lambda is diagonalizably realizable ( Dℛ {\mathcal{D {\mathcal R} }} ) if a realizing matrix A A is diagonalizable, and Λ \Lambda is universally realizable ( Uℛ {\mathcal{U {\mathcal R} }} ) if it is realizable for each Jordan canonical form allowed by Λ . \Lambda . Here, we consider the universal realizability problem for lists Λ = { λ 1 , λ 2 , … , λ n } \Lambda =\left\{{\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{n}\right\} of complex numbers in the left half-plane, that is, lists with λ 1 > 0 {\lambda }_{1}\gt 0 , Re λ i ≤ 0 {\rm{Re}}{\lambda }_{i}\le 0 , i = 2 , … , n . i=2,\ldots ,n. Then, we show that Dℛ {\mathcal{D {\mathcal R} }} implies Uℛ {\mathcal{U {\mathcal R} }} and we provide necessary and sufficient conditions under which a realizable left half-plane list of complex numbers is Uℛ {\mathcal{U {\mathcal R} }} .

Abstract Circulant matrices form an important class of matrices that have been continuously studied due to their nice algebraic structures and wide applications. In this study, we focus specifically on negacirculant matrices, which are known as extensions of the classical circulant matrices. The algebraic structures of the rings of n × n n\times n negacirculant matrices over finite fields and over finite commutative chain rings are presented. Subsequently, the algebraic structures and enumeration of the unit groups of such matrix rings are established. Additionally, the number of non-singular n × n n\times n negacirculant matrices with prescribed determinant is given in some cases. Conjectures and open problems are proposed as well as a brief discussion in the case where the underlying ring is a finite commutative ring with identity is also presented.

Abstract The class of square matrices of order n n having a positive determinant and all their minors up to order n − 1 n-1 nonpositive is considered. A characterization of these matrices based on the Cauchon algorithm is presented, which provides an easy test for their recognition. Furthermore, it is shown that all matrices lying between two matrices of this class with respect to the checkerboard ordering are contained in this class, too. For both results, we require that the entry in the bottom-right position is negative.

Abstract For the graphs G G and H H , the spectral deviation of H H from G G is defined as ϱ G ( H ) = ∑ μ ∈ H min λ ∈ G ∣ λ − μ ∣ , {\varrho }_{G}\left(H)=\sum _{\mu \in H}\mathop{\min }\limits_{\lambda \in G}| \lambda -\mu | , where ∈ \in designates that the given number is an eigenvalue of the adjacency matrix of the corresponding graph. In this study, we consider the problem of existence of a proper induced subgraph H H of a prescribed graph G G such that ϱ G ( H ) = 0 {\varrho }_{G}\left(H)=0 , and the problem of determination of all such subgraphs. We investigate these problems in the framework of Smith graphs and their induced subgraphs, graphs with small second largest eigenvalue, graphs with small number of either positive or distinct eigenvalues, integral graphs, and chain graphs. Our results can be interesting in the context of graphs with a fixed number of distinct eigenvalues, eigenvalue distribution, or spectral distances of graphs.

Abstract This study presents the matrix variate inverse Lomax distribution as a generalization of the univariate inverse Lomax distribution and shows that this distribution can be derived by using matrix variate gamma distributions. We study several properties such as cumulative distribution function, marginal distribution of sub-matrix, triangular factorization, moment generating function, Kullback-Liebler divergence, entropies and expected values of the inverse Lomax matrix. We also derive maximum likelihood estimators of the parameters and check their finite sample performance by simulations. Some of these results are expressed in terms of special functions of matrix arguments and zonal polynomials.

Abstract We explicitly determine all connected graphs whose Laplacian matrices have at most four eigenvalues different from 1 and 2.

Abstract This article considers the computation of the matrix exponential e A {{\rm{e}}}^{A} with numerical quadrature. Although several quadrature-based algorithms have been proposed, they focus on (near) Hermitian matrices. In order to deal with non-Hermitian matrices, we use another integral representation including an oscillatory term and consider applying the double exponential (DE) formula specialized to Fourier integrals. The DE formula transforms the given integral into another integral whose interval is infinite, and therefore, it is necessary to truncate the infinite interval. In this article, to utilize the DE formula, we analyze the truncation error and propose two algorithms. The first one approximates e A {{\rm{e}}}^{A} with the fixed mesh size, which is a parameter in the DE formula affecting the accuracy. The second one computes e A {{\rm{e}}}^{A} based on the first one with automatic selection of the mesh size depending on the given error tolerance.

Abstract In this article, we study eigenvalues λ \lambda and their associated eigenvectors x x of the adjacency matrices A A of balanced signed graphs. Balanced signed graphs were first introduced and studied by Harary to handle a problem in social psychology. Harary showed in 1953 that a signed graph is balanced if and only if its vertex set V V can be divided into two sets (either of which may be empty), X X and Y Y , so that each edge between the sets is negative and each within a set is positive. Based on this fundamental theorem for the balanced signed graphs, vertices of a balanced signed graph can be labeled in a way so that its adjacency matrix is well structured. Using this special structure, we find algebraically all eigenvalues and their associated eigenvectors of the adjacency matrix A A of a given balanced signed graph. We present in this study eigenpairs ( λ , x ) \left(\lambda ,x) of adjacency matrices of balanced signed graphs with some special structures.

Abstract Let G G be a simple graph with adjacency matrix A ( G ) A\left(G) , degree diagonal matrix D ( G ) , D\left(G), and let l ( G ) l\left(G) be the line graph of G G . In 2017, Nikiforov defined the A α {A}_{\alpha } -matrix of G G , A α ( G ) {A}_{\alpha }\left(G) , as a linear convex combination of A ( G ) A\left(G) and D ( G ) D\left(G) , in the following way, A α ( G ) ≔ α A ( G ) + ( 1 − α ) D ( G ) , {A}_{\alpha }\left(G):= \alpha A\left(G)+\left(1-\alpha )D\left(G), where α ∈ [ 0 , 1 ] \alpha \in \left[0,1] . In this study, we present some bounds for the largest eigenvalue of A α ( G ) , {A}_{\alpha }\left(G), and for some eigenvalues of A α ( l ( G ) ) {A}_{\alpha }\left(l\left(G)) . Extremal graphs attaining some of these bounds are also characterized. Furthermore, some comparisons between the new bounds obtained in this study, and between these and some bounds presented by Nikiforov are made.