Volume 13, Issue 1

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.

Let T n = tridiag ( − 1 , b , − 1 ) {T}_{n}={\rm{tridiag}}\left(-1,b,-1) , an n × n n\times n symmetric, strictly diagonally dominant tridiagonal matrix ( ∣ b ∣ > 2 | b| \gt 2 ). This article investigates tridiagonal near-Toeplitz matrices T ˜ n ≔ [ t ˜ i , j ] {\widetilde{T}}_{n}:= \left[{\widetilde{t}}_{i,j}] , obtained by perturbing the ( 1 , 1 ) \left(1,1) and ( n , n ) \left(n,n) entry of T n {T}_{n} . Let t ˜ 1 , 1 = t ˜ n , n = b ˜ ≠ b {\widetilde{t}}_{1,1}={\widetilde{t}}_{n,n}=\widetilde{b}\ne b . We derive exact inverses of T ˜ n {\widetilde{T}}_{n} . Furthermore, we demonstrate that these results hold even when ∣ b ˜ ∣ < 1 | \widetilde{b}| \lt 1 . Additionally, we establish upper bounds for the infinite norms of the inverse matrices. The row sums and traces of the inverse provide insight into the matrix’s spectral properties and play a key role in understanding the convergence of fixed-point iterations. These metrics allow us to derive tighter bounds on the infinite norms and improve computational efficiency. Numerical results for Fisher’s problem demonstrate that the derived bounds closely match the actual infinite norms, particularly for b > 2 b\gt 2 with b ˜ ≤ 1 \widetilde{b}\le 1 and b < − 2 b\lt -2 with b ˜ ≥ − 1 \widetilde{b}\ge -1 . For other cases, further refinement of the bounds is possible. Our results contribute to improving the convergence rates of fixed-point iterations and reducing the computation time for matrix inversion.

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.

In this article, we study the combinatorics of permutation polytopes of some cyclic groups, denoted by P ⟨ g ⟩ P\langle g\rangle where g ∈ S n g\in {S}_{n} . We give a formula on the dimension and the number of vertices of the smallest faces containing two given vertices. Then, we study the number of the smallest faces containing one fixed vertex and some other vertex which are k k -dimensional. As a consequence, the diameter of the vertex-edge graph of some special permutation polytopes P ⟨ g ⟩ P\langle g\rangle is obtained. We also consider the reduced Gröbner basis of the toric ideal of some special P ⟨ g ⟩ P\langle g\rangle and show that in this case P ⟨ g ⟩ P\langle g\rangle is compressed.

Attempts to resolve the Akbari-Cameron-Khosrovshahi-conjecture have so far focused on the rank of a matrix. The conjecture claims that there exists a nonzero (0, 1)-vector in the row space of a (0, 1)-adjacency matrix A {\bf{A}} of a graph G G , that is not a copy of any row of A {\bf{A}} . We present a new approach different from the methods used to date. By considering the change in the nullity of A {\bf{A}} on adding an arbitrary vertex to a base graph G G , we seek counter-examples to the conjecture. As a result, we determine a class C {\mathcal{C}} of graphs that could be potential counter-examples to the conjecture. We use eigenvector techniques to show that C {\mathcal{C}} is restricted to the intersection of a number of families of graphs with particular properties.

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.

For any real number α ∈ [ 0,1 ] \alpha \in \left[\mathrm{0,1}] , by the A α {A}_{\alpha } - matrix of a graph G G , we mean the matrix A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) {A}_{\alpha }\left(G)=\alpha D\left(G)+\left(1-\alpha )A\left(G) , where A ( G ) A\left(G) and D ( G ) D\left(G) are the adjacency matrix and the diagonal matrix of vertex degrees of G G , respectively. The largest eigenvalue of A α ( G ) {A}_{\alpha }\left(G) is called the A α {A}_{\alpha } -index of G G . Chang and Tam ( Graphs of fixed order and size with maximal-index , Linear Algebra Appl. 673 (2023), 69–100) have solved the problem of determining graphs with maximal A α {A}_{\alpha } -index over G ( n , m ) {\mathcal{G}}\left(n,m) , the class of graphs with n n vertices and m m edges, for α ∈ 1 2 , 1 \alpha \in \left[\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2},1\right) and 1 ≤ m ≤ 2 n − 3 1\le m\le 2n-3 . In the same article, they posed the problem of characterizing graphs in G ( n , m ) {\mathcal{G}}\left(n,m) that maximize the A α {A}_{\alpha } -index for 0 < α < 1 2 0\lt \alpha \lt \frac{1}{2} and m ≤ n − 1 m\le n-1 . In this study, it is noted that, for any α ∈ \alpha \hspace{0.33em}\in [0, 1), the problem of characterizing graphs with maximal A α {A}_{\alpha } -index over G ( n , m ) {\mathcal{G}}\left(n,m) with m ≤ n − 1 m\le n-1 is equivalent to the problem of characterizing graphs with maximal A α {A}_{\alpha } -index over S ( m ) {\mathscr{S}}\left(m) , the class of graphs with m m edges. In connection with the latter problem, we pose the following conjecture: Let m ≥ 3 m\ge 3 be a positive integer and suppose that m = d 2 + t m=\left(\phantom{\rule[-0.75em]{}{0ex}},\genfrac{}{}{0ex}{}{d}{2}\right)+t with 0 ≤ t < d 0\le t\lt d . There exists a real number α 0 {\alpha }_{0} , α 0 = 1 2 {\alpha }_{0}=\frac{1}{2} for m = 3 m=3 and α 0 ∈ [ 0 , 1 2 ) {\alpha }_{0}\in \left[0,\frac{1}{2}) for m ≥ 4 m\ge 4 , such that for any α ∈ \alpha \hspace{0.33em}\in [0, 1), C d + 1 m {C}_{d+1}^{m} (replaced by K d {K}_{d} , in case t = 0 t=0 ), where C n m {C}_{n}^{m} denotes the quasi-complete graph with n n vertices and m m edges, or K 1 , m {K}_{1,m} is the unique connected graph with m m edges that maximize the A α {A}_{\alpha } -index over S ( m ) {\mathscr{S}}\left(m) , depending on whether α ∈ [ 0 , α 0 ) \alpha \in \left[0,{\alpha }_{0}) or α ∈ ( α 0 , 1 ) \alpha \in \left({\alpha }_{0},1) ; when α = α 0 \alpha ={\alpha }_{0} , there are exactly two connected graphs that maximize the A α {A}_{\alpha } -index over S ( m ) {\mathscr{S}}\left(m) , namely, C d + 1 m {C}_{d+1}^{m} (or K d {K}_{d} , in case t = 0 t=0 ) and K 1 , m {K}_{1,m} . The conjecture is established when t = 0 t=0 .

The generalized distance matrix of a graph is a matrix in which the ( i , j ) \left(i,j) th entry is a function, f f , of the distance between vertex i i and vertex j j . Depending on the choice of f f , this family of matrices includes both the adjacency matrix and the traditional distance matrix. We present a cospectral construction for the generalized distance matrix akin to Godsil-McKay Switching. We also investigate a special case of the generalized distance matrix: the exponential distance matrix, which is a matrix where every entry is a value q q raised to the power of the distance between the vertices. We give an upper bound on the values of q q needed to show a pair of graphs is cospectral for all values of q q corresponding to the diameter of the graphs. We also give cospectral constructions unique to value q = 1 ⁄ 2 q=1/2 .

The generalized eigenvalue problem is a significant topic with numerous applications in scientific and technological computing. Therefore, verifying the reliability of the results produced by numerical solvers is crucial. This article presents a method for generating test problems for generalized eigenvalue problems such that the exact eigenvalues and eigenvectors are known in advance, thereby enabling precise error monitoring. The proposed method generates a matrix pair that can be represented exactly in floating-point numbers. Ozaki and Ogita previously proposed a method for generating test problems for standard eigenvalue problems. We extend their approach to generalized eigenvalue problems, proposing a method that allows users to specify target eigenvalues and a condition number of the matrix defining the generalized inner product in advance. Furthermore, iterative refinement methods to approach the expected eigenvalues as closely as possible for the user are also discussed. Numerical examples are presented to evaluate the accuracy of the computed results and to assess potential overestimation in the error bounds produced by a method for given approximate eigenvalues.

We establish tight lower bounds for the trace norm ( ‖ ⋅ ‖ 1 ) \left(\Vert \cdot {\Vert }_{1}) of real symmetric and Hermitian matrices with zero diagonal entries in terms of their entrywise L 1 {L}^{1} -norms ( ‖ ⋅ ‖ ( 1 ) ) \left(\Vert \cdot {\Vert }_{\left(1)}) . For the space of nonzero real symmetric matrices of order n n , we prove that the minimum possible ratio ‖ A ‖ 1 ‖ A ‖ ( 1 ) \frac{\Vert A{\Vert }_{1}}{\Vert A{\Vert }_{\left(1)}} is exactly 2 n \frac{2}{n} . In the Hermitian case, this minimum ratio is given by tan π 2 n \tan \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{\pi }{2n}\right) . Through duality arguments, we derive sharp upper bounds for the spectral norm distance between a matrix and the space of diagonal matrices. For instance, any real symmetric matrix with off-diagonal entries bounded by 1 in absolute value lies within a spectral norm distance of n 2 \frac{n}{2} from a diagonal matrix, while the corresponding bound for Hermitian matrices is cot π 2 n \cot \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{\pi }{2n}\right) . Applications to graph energy and quantum coherence are discussed, highlighting implications for algebraic graph theory and quantum resource theory.

In this article, we study the block diagonalization of ( p , q ) \left(p,q) -tridiagonal matrices and derive closed-form expressions for the number and structure of diagonal blocks as functions of the parameters p p , q q , and n n . This reduction enables efficient computation of eigenvalues and eigenvectors by decomposing the matrix into smaller subproblems. We extend the method to more general ( P , Q ) \left({\mathcal{P}},{\mathcal{Q}}) -tridiagonal matrices, where P {\mathcal{P}} and Q {\mathcal{Q}} are sets of positive integers, covering general banded structures. We also examine special cases such as bidiagonal and triangular block reductions along with supporting algorithms and numerical examples.

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.

We show how to obtain analytic expressions for the Jordan form of an upper triangular matrix, including that of the standardizing matrix. For example, the Jordan form of a triangular banded matrix (a banded matrix is a matrix where nonzero entries are confined to a band along the main diagonal) is given in terms of Bell polynomials.

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} }} .

This study establishes lower and upper bounds for the eigenvalues of a symmetric pentadiagonal matrix arising in the Hodrick-Prescott (HP) filter, a widely used method for trend extraction in macroeconometrics. The derivation relies on the Cauchy interlacing theorem. Based on the results, we further derive bounds for the eigenvalues of the smoother matrix of the HP filter. Additionally, we present several related findings that complement our main results.

A nonzero element a a is called 1-Sylvester in a ring R R , if there exist b , c ∈ R b,c\in R such that 1 = a b + c a 1=ab+ca . In this article, we study such elements, mainly in matrix rings over commutative rings. In particular, we study the case when b = c b=c , when b b is called an anticommutator inverse for a a .

Let G G be a simple graph on n n vertices with degree sequence d 1 , … , d n {d}_{1},\ldots ,{d}_{n} . Fajtlowicz ( On conjectures of Graffiti , Discrete Math. 72 (1988), 113–118) defined the temperature of a vertex v v of G G as d n − d \frac{d}{n-d} , where d d is the degree of v v . Motivated by this definition, we define the temperature index of G G , denoted by T ( G ) T\left(G) , as T ( G ) = d 1 n − d 1 + ⋯ + d n n − d n T\left(G)=\frac{{d}_{1}}{n-{d}_{1}}+\cdots +\frac{{d}_{n}}{n-{d}_{n}} . We obtain some lower bounds and upper bounds for T ( G ) T\left(G) in terms of the number of vertices, the number of edges, the maximum and the minimum vertex degree and the Zagreb index ( Z ( G ) = d 1 2 + ⋯ + d n 2 Z\left(G)={d}_{1}^{2}+\cdots +{d}_{n}^{2} ). Using these results we derive new bounds for the Zagreb index of graphs. Finally, we study the temperature index of graphs from the point of view of spectra of graphs (the eigenvalues of their adjacency matrices). In particular, we show that G G has at least one eigenvalue in the interval [ − n − δ T ( G ) − 2 m n , n − δ T ( G ) − 2 m n ] \left[-\sqrt{n-\delta }\sqrt{T\left(G)-\frac{2m}{n}},\sqrt{n-\delta }\sqrt{T\left(G)-\frac{2m}{n}}] .

We introduce an operator-theoretic definition of a proper graph Laplacian as any matrix associated with a given graph that can be expressed as the composition of a divergence and a gradient operator, with the gradient acting between graph-related spaces and annihilating constant functions. This provides a unified framework for determining whether a matrix represents a genuine diffusive operator on a graph. Within this framework, we prove that the standard Laplacian, the fractional Laplacian, the d d -path Laplacians, and the degree-attracting and degree-repelling Laplacians are all proper diffusive Laplacians . In contrast, the in-degree and out-degree Laplacians correspond to advection operators, while the signed, signless, magnetic, and deformed Laplacians are improper , as they cannot be written as the composition of a divergence and a true gradient. The magnetic Laplacian is shown to arise as the Schur complement of an extended proper Laplacian defined on a higher-dimensional space, a property also inherited by the signless Laplacian. The Lerman-Ghosh Laplacian is identified as a nonconservative diffusive operator coupled to an external reservoir. Finally, we prove that the Moore-Penrose pseudoinverse of the Laplacian is itself a proper Laplacian. This classification establishes a rigorous operator-theoretic foundation for distinguishing proper, nonconservative, and improper graph Laplacians.