Volume 12, Issue 1

The geometry of the compact convex set of all n × n n\times n doubly stochastic matrices, a structure frequently referred to as the Birkhoff polytope, has been an active subject of research as of late. Geometric characteristics such as the Chebyshev center and the Chebyshev radius with respect to the operator norms from ℓ n p {\ell }_{n}^{p} to ℓ n p {\ell }_{n}^{p} and the Schatten p p -norms, both for the range 1 ≤ p ≤ ∞ 1\le p\le \infty , have only recently been studied in depth. In this article, we continue in this vein by determining the diameter of the Birkhoff polytope with respect to the metrics induced by the aforementioned matrix norms.

Diagonal matrices and their generalization in terms of tridiagonal matrices have been of interest due to their nice algebraic properties and wide applications. In this article, the determinants of tridiagonal matrices over a finite field F q {{\mathbb{F}}}_{q} and a commutative finite chain ring R R are studied. The main focus is the enumeration of tridiagonal matrices with prescribed determinant. The number of tridiagonal matrices with prescribed determinant over F q {{\mathbb{F}}}_{q} and the number of non-singular tridiagonal matrices with prescribed determinant over R R are completely determined. For singular tridiagonal matrices with prescribed determinant over R R , bounds on the number of such matrices with prescribed determinant are given. Subsequently, the number of some special tridiagonal matrices with prescribed determinant over F q {{\mathbb{F}}}_{q} and R R is presented.

Let A A be an arbitrary matrix in which the number of rows, m m , is considerably larger than the number of columns, n n . Let the submatrix A i , i = 1 , … , m {A}_{i},\hspace{0.33em}i=1,\ldots ,m , be composed from the first i i rows of A A , and let β i {\beta }_{i} denote the smallest singular value of A i {A}_{i} . Recently, we observed that the first part of this sequence, β 1 , … , β n {\beta }_{1},\ldots ,{\beta }_{n} , is descending, while the second part, β n , … , β m {\beta }_{n},\ldots ,{\beta }_{m} , is ascending. This property is called “the smallest singular value anomaly.” In this article, we expose another interesting feature of this sequence. It is shown that certain types of matrices possess the sharp anomaly phenomenon: First, when i i is considerably smaller than n n , the value of β i {\beta }_{i} decreases rather slowly. Then, as i i approaches n n from below, there is fast reduction in the value of β i {\beta }_{i} , making β n {\beta }_{n} much smaller than β 1 {\beta }_{1} . Yet, once i i passes n n , the situation is reversed and β i {\beta }_{i} increases rapidly. Finally, when i i moves away from n n , the rate of increase slows down. The article illustrates this behavior and explores its reasons. It is shown that the sharp anomaly phenomenon occurs in matrices with “scattering rows.”

Over any GCD (greatest common divisor) commutative domain we show that the nontrivial 2 × 2 2\times 2 idempotent matrices are products of two nilpotent matrices. In order to find explicitly such decompositions, two procedures are described. Assisted by computer, we were able to find an example of idempotent 2 × 2 2\times 2 matrix over Z [ − 5 ] {\mathbb{Z}}\left[\sqrt{-5}] , which shows that the GCD condition is (sufficient but) not necessary. Finally, a generalization is discussed and some open questions stated.

A family of two-unitary complex Hadamard matrices (CHMs) of size 36 stemming from a particular matrix is constructed. Every matrix in this orbit remains unitary after operations of partial transpose and reshuffling which makes it a distinguished subset of CHM. It provides a novel solution to the quantum version of the Euler problem, in which each field of the Graeco-Latin square of size six contains a symmetric superposition of all 36 officers with phases being multiples of sixth root of unity. This simplifies previously known solutions as all amplitudes of the superposition are equal and the set of phases consists of six elements only. Multidimensional parameterization allows for more flexibility in a potential experimental realization.

Let Ω n {\Omega }_{n} denote the set of all doubly stochastic matrices of order n n . Lih and Wang conjectured that for n ≥ 3 n\ge 3 , per ( t J n + ( 1 − t ) A ) ≤ t \left(t{J}_{n}+\left(1-t)A)\le t per J n + ( 1 − t ) {J}_{n}+\left(1-t) per A A , for all A ∈ Ω n A\in {\Omega }_{n} and all t ∈ [ 0.5 , 1 ] t\in \left[0.5,1] , where J n {J}_{n} is the n × n n\times n matrix with each entry equal to 1 n \frac{1}{n} . This conjecture was proved partially for n ≤ 5 n\le 5 . Let K n {K}_{n} denote the set of nonnegative n × n n\times n matrices whose elements have sum n n . Let ϕ \phi be a real valued function defined on K n {K}_{n} by ϕ ( X ) = ∏ i = 1 n r i + ∏ j = 1 n c j \phi \left(X)={\prod }_{i=1}^{n}{r}_{i}+{\prod }_{j=1}^{n}{c}_{j} - per X X for X ∈ K n X\in {K}_{n} with row sum vector ( r 1 , r 2 , … r n ) \left({r}_{1},{r}_{2},\ldots {r}_{n}) and column sum vector ( c 1 , c 2 , … c n ) \left({c}_{1},{c}_{2},\ldots {c}_{n}) . A matrix A ∈ K n A\in {K}_{n} is called a ϕ \phi -maximizing matrix if ϕ ( A ) ≥ ϕ ( X ) \phi \left(A)\ge \phi \left(X) for all X ∈ K n X\in {K}_{n} . Dittert conjectured that J n {J}_{n} is the unique ϕ \phi -maximizing matrix on K n {K}_{n} . Sinkhorn proved the conjecture for n = 2 n=2 and Hwang proved it for n = 3 n=3 . In this article, we prove the Lih and Wang partially for n = 6 n=6 . It is also proved that if A A is a ϕ \phi -maximizing matrix on K 4 {K}_{4} , then A A is fully indecomposable.

In this article, we establish a new closed formula for the solution of homogeneous second-order linear difference equations with constant coefficients by using matrix theory. This, in turn, gives new closed formulas concerning all sequences of this type such as the Fibonacci and Lucas sequences. Next, we show the main advantage of our formula which is based on the fact that future calculations rely on the previous ones and this is true from any desired starting point. As applications, we show that Binet’s formula, in this case, is valid for negative integers as well. Finally, new summation formulas relating elements of such sequences. As a conclusion, we present a formula for the sum of squares of Chebychev polynomials of the first and second kind.

We consider the convolution equation F * X = B F* X=B , where F ∈ R 3 × 3 F\in {{\mathbb{R}}}^{3\times 3} and B ∈ R m × n B\in {{\mathbb{R}}}^{m\times n} are given and X ∈ R m × n X\in {{\mathbb{R}}}^{m\times n} is to be determined. The convolution equation can be regarded as a linear system with a coefficient matrix of special structure. This fact has led to many studies including efficient numerical algorithms for solving the convolution equation. In this study, we show that the convolution equation can be represented as a generalized Sylvester equation. Furthermore, for some realistic examples arising from image processing, we show that the generalized Sylvester equation can be reduced to a simpler form, and we analyze the unique solvability of the convolution equation.

Given matrices A A and B B of the same order, A A is called a section of B B if R ( A ) ∩ R ( B − A ) = { 0 } {\mathscr{R}}\left(A)\cap {\mathscr{R}}\left(B-A)=\left\{0\right\} and R ( A T ) ∩ R ( ( B − A ) T ) = { 0 } {\mathscr{R}}\left({A}^{T})\cap {\mathscr{R}}\left({\left(B-A)}^{T})=\left\{0\right\} , where R ( . ) {\mathscr{R}}\left(.) denotes the range (column space) of a matrix argument and the superscript T T stands for the transpose of a matrix. A matrix G G is called positive semidefinite (p.s.d.) if x T G x {x}^{T}Gx is nonnegative for all real vector x x . However, we refer to a symmetric p.s.d. matrix simply as a p.s.d. matrix as the applications discussed in this article are concerned with only symmetric p.s.d. matrices. An n × n n\times n p.s.d. matrix G G admits of a symmetric section G X {G}_{X} of G G with regard to an n × k n\times k matrix X X such that R ( G X ) = R ( G ) ∩ R ( X ) {\mathscr{R}}\left({G}_{X})={\mathscr{R}}\left(G)\cap {\mathscr{R}}\left(X) . In this article, sections of the type G X {G}_{X} are used in minimization of quadratic functions under linear constraints and splitting of vector random variables into uncorrelated vector random variables. In the general Gauss-Markoff model, y = X β + ε y=X\beta +\varepsilon , with design matrix X X and a singular covariance matrix σ 2 G {\sigma }^{2}G of ε \varepsilon , y y is decomposed into four uncorrelated vector random variables as y = M 1 y + M 2 y + M 3 y + M 4 y y={M}_{1}y+{M}_{2}y+{M}_{3}y+{M}_{4}y , where M i , i = 1 , 2 , 3 {M}_{i},i=1,2,3 consist of sections of G G and X X T X{X}^{T} , and M 4 {M}_{4} is a matrix whose row space is the null space of G + X X T G+X{X}^{T} .

We find the spectrum and eigenvectors of an arbitrary irreducible complex tridiagonal matrix with two-periodic main diagonal. This is expressed in terms of the spectrum and eigenvectors of the matrix with the same sub- and superdiagonals and zero main diagonal. Our result generalises some recent results where the latter matrix stemmed from certain discrete orthogonal polynomials including specific cases of the classical Krawtchouk and Hahn polynomials.

Let G G be a graph. Then, the inverse graph G − 1 {G}^{-1} of G G is defined to be a graph that has adjacency matrix similar to the inverse of the adjacency matrix of G G , where the similarity matrix is ± 1 \pm 1 diagonal matrix. In this article, we introduced a generalization of this definition that serves the mixed graphs where the definition applied for the α \alpha -Hermitian adjacency matrices of mixed graphs. Furthermore, for a class of unicyclic graphs, we were able to find an inverse mixed graph for a graph G G , where it was proven that G − 1 {G}^{-1} does not exist.

Let G G be a connected graph with n n vertices v 1 , … , v n {v}_{1},\ldots ,{v}_{n} . The Harary matrix of G G , denoted by H ( G ) H\left(G) , is an n × n n\times n matrix with a zero main diagonal, where the ( i , j ) \left(i,j) -entry is 1 d ( v i , v j ) \frac{1}{d\left({v}_{i},{v}_{j})} for i ≠ j i\ne j , and d ( v i , v j ) d\left({v}_{i},{v}_{j}) represents the distance between v i {v}_{i} and v j {v}_{j} . Let ρ 1 , … , ρ n {\rho }_{1},\ldots ,{\rho }_{n} be the eigenvalues of H ( G ) H\left(G) . The Harary Estrada index of G G is defined as HEE ( G ) = ∑ i = 1 n e ρ i {\rm{HEE}}\left(G)=\mathop{\sum }\limits_{i=1}^{n}{e}^{{\rho }_{i}} . In this article we study Harary Estrada index and find some new sharp bounds for the Harary Estrada index of graphs. Our results generalize and improve the previous bounds on the Harary Estrada index of graphs. Finally, we obtain some Nordhaus-Gaddum type inequalities for the Harary Estrada index of graphs.

The Palais matrix represents an n n -dimensional rotation between two vectors that is functionally equivalent to the Householder reflection. This study introduces a one-parameter family of unitary transforms, termed the θ \theta transform, which encompasses the transform by the Palais matrix, the Householder reflection, and their unitary extensions. Furthermore, we define the θ ∠ {\theta }_{\angle } transform , a variant of the θ \theta transform featuring bounded component norms. It is demonstrated that the θ ∠ {\theta }_{\angle } transform is computationally efficient and backward stable when one of the vectors has the “one-hot” structure, making it highly valuable for matrix decompositions such as the QR decomposition. In addition, the θ \theta transform exhibits additional characteristics, including its convergence to the identity and the rowwise structure of its backward error.

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.

We investigate refined inertias of positive patterns and patterns that have each off-diagonal entry positive and each diagonal entry zero, i.e., hollow positive patterns. For positive patterns, we prove that every refined inertia ( n + , n − , n z , 2 n p ) \left({n}_{+},{n}_{-},{n}_{z},2{n}_{p}) with n + ≥ 1 {n}_{+}\ge 1 can be realized. For hollow positive patterns, we prove that every refined inertia with n + ≥ 1 {n}_{+}\ge 1 and n − ≥ 2 {n}_{-}\ge 2 can be realized. To illustrate these results, we construct matrix realizations using circulant matrices and bordered circulants. For both patterns of order n n , we show that as n → ∞ n\to \infty , the fraction of possible refined inertias realized by circulants approaches 1/4 for n n odd and 3/4 for n n even.

In this study, we derive the Drazin inverse ( A + ε B ) D {\left(A+\varepsilon B)}^{D} of the complex matrix A + ε B A+\varepsilon B with Ind ( A + ε B ) > 1 {\rm{Ind}}\left(A+\varepsilon B)\gt 1 and Ind ( A ) = k {\rm{Ind}}\left(A)=k and the group inverse ( A + ε B ) # {\left(A+\varepsilon B)}^{\#} of the complex matrix A + ε B A+\varepsilon B with Ind ( A + ε B ) = 1 {\rm{Ind}}\left(A+\varepsilon B)=1 and Ind ( A ) = k {\rm{Ind}}\left(A)=k when ε B \varepsilon B is viewed as the perturbation of A A . If the dual Drazin inverse (DDGI) A ^ DDGI {\widehat{A}}^{{\rm{DDGI}}} of A ^ \widehat{A} is considered as a notation. We calculate ( A + ε B ) D − A ^ DDGI {\left(A+\varepsilon B)}^{D}-{\widehat{A}}^{{\rm{DDGI}}} and ( A + ε B ) # − A ^ DDGI {\left(A+\varepsilon B)}^{\#}-{\widehat{A}}^{{\rm{DDGI}}} and obtain ‖ ( A + ε B ) D − A ^ DDGI ‖ P ∈ O ( ε 2 ) \Vert {\left(A+\varepsilon B)}^{D}-{\widehat{A}}^{{\rm{DDGI}}}{\Vert }_{P}\in {\mathcal{O}}\left({\varepsilon }^{2}) and ‖ ( A + ε B ) # − A ^ DDGI ‖ P ∈ O ( ε 2 ) \Vert {\left(A+\varepsilon B)}^{\#}-{\widehat{A}}^{{\rm{DDGI}}}{\Vert }_{P}\in {\mathcal{O}}\left({\varepsilon }^{2}) . Meanwhile, we give some examples to verify these conclusions.

Let G = ( V ( G ) , E ( G ) ) G=\left(V\left(G),E\left(G)) be a graph of order n n . The exponential atom-bond connectivity matrix A e ABC ( G ) {A}_{{e}^{{\rm{ABC}}}}\left(G) of G G is an n × n n\times n matrix whose ( i , j ) \left(i,j) -entry is equal to e d ( v i ) + d ( v j ) − 2 d ( v i ) d ( v j ) {e}^{\sqrt{\tfrac{d\left({v}_{i})+d\left({v}_{j})-2}{d\left({v}_{i})d\left({v}_{j})}}} if v i v j ∈ E ( G ) {v}_{i}{v}_{j}\in E\left(G) , and 0 otherwise. The exponential atom-bond connectivity energy of G G is the sum of the absolute values of all eigenvalues of the matrix A e ABC ( G ) {A}_{{e}^{{\rm{ABC}}}}\left(G) . It is proved that among all trees of order n n , the star S n {S}_{n} is the unique tree with the minimum exponential atom-bond connectivity energy.

It is known that the inverse of an invertible real square matrix satisfying the triangle property is a tridiagonal matrix. In this note, results that may be considered as analogues of this assertion are obtained for the Moore-Penrose inverse and the group inverse.

A simple proof using Birkhoff theorem is given for the result that the spectral norm of a doubly stochastic matrix is 1. We also show that the result generalizes the results of İpek, Bozkurt, and Jiang and Zhou on circulant matrices and r r -circulant matrices. Spectral norm of level- k k circulant matrix and applications are given.

We consider two types of joins of graphs G 1 {G}_{1} and G 2 {G}_{2} , G 1 ⊻ G 2 {G}_{1}\hspace{0.33em}⊻\hspace{0.33em}{G}_{2} – the neighbors splitting join and G 1 ∨ = G 2 {G}_{1}\mathop{\vee }\limits_{=}{G}_{2} – the nonneighbors splitting join, and compute the adjacency characteristic polynomial, the Laplacian characteristic polynomial, and the signless Laplacian characteristic polynomial of these joins. When G 1 {G}_{1} and G 2 {G}_{2} are regular, we compute the adjacency spectrum, the Laplacian spectrum, the signless Laplacian spectrum of G 1 ∨ = G 2 {G}_{1}\mathop{\vee }\limits_{=}{G}_{2} , and the normalized Laplacian spectrum of G 1 ⊻ G 2 {G}_{1}\hspace{0.33em}⊻\hspace{0.33em}{G}_{2} and G 1 ∨ = G 2 {G}_{1}\mathop{\vee }\limits_{=}{G}_{2} . We use these results to construct nonregular, nonisomorphic graphs that are cospectral with respect to the four matrices: adjacency, Laplacian, signless Laplacian and normalized Laplacian.

The sub-defect of a doubly stochastic matrix A A , denoted as s d ( A ) = ⌈ n − sum ( A ) ⌉ sd\left(A)=\lceil n-{\rm{sum}}\left(A)\rceil , is defined as the minimum number of rows and columns required to be added to transform the doubly substochastic matrix into a doubly stochastic matrix. Here, n n signifies the matrix size, and sum ( A ) {\rm{sum}}\left(A) represents the sum of all entries of A A . This article systematically examines the sub-defect characteristics inherited in doubly stochastic matrices, specifically in the context of symmetric, Hankel-symmetric, and centrosymmetric doubly substochastic matrices. Furthermore, we present illustrative examples to elucidate the practical applicability and significance of our approach in comprehending and manipulating the sub-defect of these specialized matrices.

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.

A complex unit gain graph (or T {\mathbb{T}} -gain graph) Γ = ( G , γ ) \Gamma =\left(G,\gamma ) is a gain graph with gains in T {\mathbb{T}} , the multiplicative group of complex units. The T {\mathbb{T}} -outgain in Γ \Gamma of a vertex v ∈ G v\in G is the sum of the gains of all the arcs originating in v v . A T {\mathbb{T}} -gain graph is said to be an a a - T {\mathbb{T}} -regular graph if the T {\mathbb{T}} -outgain of each of its vertices is equal to a a . In this article, it is proved that a a - T {\mathbb{T}} -regular graphs exist for every a ∈ R a\in {\mathbb{R}} . This, in particular, means that every real number can be a T {\mathbb{T}} -gain graph eigenvalue. Moreover, denoted by Ω ( a ) \Omega \left(a) the class of connected T {\mathbb{T}} -gain graphs whose largest eigenvalue is the real number a a , it is shown that Ω ( a ) \Omega \left(a) is nonempty if and only if a a belongs to { 0 } ∪ [ 1 , + ∞ ) \left\{0\right\}\cup \left[1,+\infty ) . In order to achieve these results, non-complete extended p p -sums and suitably defined joins of T {\mathbb{T}} -gain graphs are considered.

In this note, we use eigenvalue interlacing to derive an inequality between a graph’s maximum degree and its maximum and minimum adjacency eigenvalues. The equality case is fully characterized.

Formulas for the determinant of distance matrix D T {D}_{T} of tree T T are known in the unweighted case and in the case when the edges of T T have commuting variable weights. Associated with the four-point condition (4PC) and a tree T T are two matrices, the Max4PC T {{\rm{Max4PC}}}_{T} and the Min4PC T {{\rm{Min4PC}}}_{T} . These are not full rank matrices and their rank, a basis B B , and formulas for the determinant when restricted to the rows and columns of B B are known. In this work, we generalize both these matrices to the case when the edges of T T have commuting variable weights and determine edge-weighted counterparts of known results.

In this article, we study the Laplacian index of tadpole graphs, which are unicyclic graphs formed by adding an edge between a cycle C k {C}_{k} and a path P n {P}_{n} . Using two different approaches, we show that their Laplacian index converges to Δ 2 Δ − 1 = 9 2 \frac{{\Delta }^{2}}{\Delta -1}=\frac{9}{2} as n , k → ∞ n,k\to \infty , where Δ = 3 \Delta =3 is the maximum degree of the graph. This limit is known as the Hoffman’s limit for the Laplacian matrix. The first technique is a linear time algorithm presented in [R. Braga, V. Rodrigues, and R. Silva, Locating eigenvalues of a symmetric matrix whose graph is unicyclic , Trends in Comput. Appl. Math. 22 (2021), no. 4, 659–674] that diagonalizes the matrix preserving its inertia. Here, we adapt this algorithm to the Laplacian index of a tadpole graph. The second method is to apply a formula presented in [V. Trevisan and E. R. Oliveira, Applications of rational difference equations to spectra graph theory , J. Difference Equ. Appl. 27 (2021), 1024–1051] for solving rational difference equations that appear when applying the diagonalization algorithm in some cases.

A (signed) graph is said to exhibit the strong reciprocal (anti-reciprocal) eigenvalue property (SR) (resp., (-SR)) if for any eigenvalue λ \lambda , it has 1 λ \frac{1}{\lambda } (resp., − 1 λ -\frac{1}{\lambda } ) as an eigenvalue as well, with the same multiplicity. It is well known that the corona of a (signed) graph does have the property -SR, and if the graph has symmetric spectrum, then it also has the property SR. Therefore, it is interesting to identify (signed) graphs which are not corona graphs with the properties SR or -SR. Recently, a few constructions for unsigned graphs with the property -SR have been offered. In this article, we extend such constructions to signed graphs.

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.

Graham and Pollak showed in 1971 that the determinant of a tree’s distance matrix depends only on its number of vertices, and, in particular, it is always nonzero. The Steiner distance of a collection of k k vertices in a graph is the fewest number of edges in any connected subgraph containing those vertices; for k = 2 k=2 , this reduces to the ordinary definition of graphical distance. Here we show that the hyperdeterminant of the k k th order Steiner distance hypermatrix is always nonzero if k k is even, extending their result beyond k = 2 k=2 . Previously, the authors showed that the k k -Steiner distance hyperdeterminant is always zero for k k odd, so together this provides a generalization to all k k . We conjecture that not just the vanishing, but the value itself, of the k k -Steiner distance hyperdeterminant of an n n -vertex tree depends only on k k and n n .

For a matrix M ∈ K n × n {\bf{M}}\in {{\mathbb{K}}}^{n\times n} , we establish a condition on the Galois group of the characteristic polynomial φ M {\varphi }_{{\bf{M}}} that induces nonvanishing of the minors of the eigenvector matrix of M {\bf{M}} . For integer matrices, recent results by Eberhard show that, conditionally on the extended Riemann hypothesis, this condition is satisfied with high probability (We say “with high probability” for probability 1 − o ( 1 ) 1-o\left(1) as n → ∞ n\to \infty .) and hence, with high probability, the minors of eigenvector matrices of random integer matrices are nonzero. For random graphs, this yields a novel uncertainty principle, related to Chebotarëv’s theorem on the roots of unity and results from Tao and Meshulam. We also show the application in graph signal processing and the connection to the rank of the walk matrix.

In this article, we give an O ( n ) O\left(n) time and space algorithm for obtaining the Laplacian eigenvalues of a cograph. This approach is more efficient as there is no need to directly compute the eigenvalues of Laplacian matrix related to this class of graphs. As an application, we use this algorithm as a tool for obtaining a closed formula for the number of spanning trees of a cograph.

Quantum walks on graphs are fundamental to quantum computing and have led to many interesting open problems in algebraic graph theory. This review article highlights three key classes of open problems in this domain: perfect state transfer, instantaneous uniform mixing, and average mixing matrices. In highlighting these open problems, our aim is to stimulate further research and exploration in this rapidly evolving field.

In this article, we study a question of Hong from 1993 related to the minimum spectral radii of the adjacency matrices of connected graphs of given order and size. Hong asked if it is true that among all connected graphs of given number of vertices n n and number of edges e e , the graphs having minimum spectral radius (the minimizer graphs) must be almost regular, meaning that the difference between their maximum degree and their minimum degree is at most one. In this article, we answer Hong’s question positively for various values of n n and e e , and in several cases, we determined the graphs with minimum spectral radius.

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

Let G G be a graph on n n vertices. In this article, we prove that an eigenbasis of the Laplacian matrix of a star graph of order n n is also an eigenbasis of G G if and only if G G is a threshold graph. As an application of this spectral characterization, we show an infinite family of threshold graphs that are weakly Hadamard diagonalizable.