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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]

Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular complex matrices or quasi-upper-triangular real matrices that are equivalent to the original matrices via unitary or, respectively, orthogonal transformations. In general, for theoretical and numerical purposes we often need to reduce, by admissible transformations, a collection of matrices to the Schur form. Unfortunately, such a reduction is not always possible. In this paper we describe all collections of complex (real) matrices that can be reduced to the Schur form by the corresponding unitary (orthogonal) transformations and explain how such a reduction can be done. We prove that this class consists of the collections of matrices associated with pseudoforest graphs. In other words, we describe when the Schur form of a collection of matrices exists and how to find it. [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]

Rayleigh–Ritz projection is one of the efficient procedures to project a large-scale quadratic eigenvalue problem (QEP) into a small-scale QEP by properly choosing a low-dimensional subspace. One common way for solving the projected QEP is to recast it via linearization. In this paper, we establish bounds for backward errors of approximate eigenpairs of QEP relative to those of a linearization. These bounds give useful information to predict the numerical stability of Rayleigh–Ritz projection eigensolvers followed by a linearization. We present results of numerical experiments that support the predictions of the backward error analysis. [ABSTRACT FROM AUTHOR]

Let Γ be a finite simple graph with adjacency matrix A. Γ is said to be singular if and only if A has an eigenvalue 0. The nullity of Γ, denoted by null $ (\Gamma) $ (Γ) , is the multiplicity of the eigenvalue 0 in the spectrum of Γ. In this paper, we completely determine the singularity of graphs Γ for which the dicylic or semi-dihedral group acts transitively and faithfully on V as a group of automorphisms. Moreover, we give formulas to calculate the nullities for such graphs. [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]

Given an arbitrary connected graph G, the n-polygon graph $ \tau _n(G) $ τ n (G) is obtained by adding a path with length n $ (n\geq ~2) $ (n ≥ 2) to each edge of graph G, and the iterated n-polygon graphs $ \tau _n^g(G) $ τ n g (G) ( $ g\geq ~0 $ g ≥ 0) are obtained from the iteration $ \tau _n^g(G)=\tau _n(\tau _n^{g-1}(G)) $ τ n g (G) = τ n (τ n g − 1 (G)) , with the initial condition $ \tau _n^0(G)=G $ τ n 0 (G) = G. In this paper, a method for calculating the eigenvalues of the normalized Laplacian matrix for graph $ \tau _n(G) $ τ n (G) is presented if the eigenvalues of a normalized Laplacian matrix for graph G is first given. The normalized Laplacian spectra for the graph $ \tau _n(G) $ τ n (G) and graphs $ \tau _n^g(G) $ τ n g (G) ( $ g\geq ~0 $ g ≥ 0) can also then be derived. Finally, as applications, we calculate the multiplicative degree-Kirchhoff index, Kemeny's constant, and the number of spanning trees for the graph $ \tau _n(G) $ τ n (G) and graphs $ \tau _n^g(G) $ τ n g (G) by exploring their connections with the normalized Laplacian spectrum, and obtain exact results for these quantities. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n. For $ \alpha \in [0,1] $ α ∈ [ 0 , 1 ] , the $ A_{\alpha } $ A α -matrix of G is defined as $ A_{\alpha }(G)=\alpha D(G)+ (1-\alpha)A(G) $ A α (G) = α D (G) + (1 − α) A (G) , where $ A(G) $ A (G) and $ D(G) $ D (G) are the adjacency matrix and the diagonal degree matrix of G, respectively. The eigenvalues of the $ A_{\alpha } $ A α -matrix of G are called the $ A_{\alpha } $ A α -eigenvalues of G. In this paper, we first study the properties on $ A_{\alpha } $ A α -eigenvalues, i.e. how the $ A_{\alpha } $ A α -eigenvalues behave under some kinds of graph transformations including vertex deletion, vertex contraction, edge deletion and edge subdivision. Moreover, we also present the relationships between the $ A_{\alpha } $ A α -eigenvalues of G and its k-domination number, independence number, chromatic number and circumference, respectively. [ABSTRACT FROM AUTHOR]

Let $ m_GI $ m G I denote the number of Laplacian eigenvalues of a graph G in an interval I and let $ \alpha (G) $ α (G) denote the independence number of G. In this paper, we determine the classes of graphs that satisfy the condition $ m_G[0,n-\alpha (G)]=\alpha (G) $ m G [ 0 , n − α (G) ] = α (G) when $ \alpha (G)= 2 $ α (G) = 2 and $ \alpha (G)= n-2 $ α (G) = n − 2 , where n is the order of G. When $ \alpha (G)=2 $ α (G) = 2 , $ G \cong K_1 \nabla K_{n-m} \nabla K_{m-1} $ G ≅ K 1 ∇ K n − m ∇ K m − 1 for some $ m \geq 2 $ m ≥ 2. When $ \alpha (G)=n-2 $ α (G) = n − 2 , there are two types of graphs $ B(p,q,r) $ B (p , q , r) and $ B'(p,q,r) $ B ′ (p , q , r) of order n = p + q + r + 2, which we call the binary star graphs. Also, we show that the binary star graphs with p = r are determined by their Laplacian spectra. [ABSTRACT FROM AUTHOR]

Let $ \mathbb {T}_4=\{1,-1,\mathbf {i},-\mathbf {i}\} $ T 4 = { 1 , − 1 , i , − i } be the group of fourth roots of unit. A $ \mathbb {T}_4 $ T 4 -gain graph is a graph where each orientation of an edge is given a complex unit in $ \mathbb {T}_4 $ T 4 , which is the inverse of the complex unit assigned to the opposite orientation. In this paper, we characterize the structure of the $ \mathbb {T}_4 $ T 4 -gain graphs with exactly one positive eigenvalue and determine the $ \mathbb {T}_4 $ T 4 -gain graphs with cut vertices having exactly two positive eigenvalues. Our results extend some parallel ones about mixed graphs and signed graphs. [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 study certain determinants involving tangent functions. We prove that for any odd number n ⩾ 3 and s ∈ Z + , the determinant D (n , s) = det (tan s ⁡ π j + k n) 1 ⩽ j , k ⩽ n − 1 ∈ n n − 2 Z . In the special case s = 2, we confirm a conjecture raised by Zhi-Wei Sun in 2019. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with a model updating problem for damped structural systems, which can be described as the following inverse quadratic eigenvalue problem (IQEP) and an optimal approximate problem (OAP). Problem IQEP: Given matrices Λ = diag { λ 1 , ... , λ p } ∈ C p × p and X = [ x 1 , ... , x p ] ∈ C n × p with p ≤ n , λ i ≠ λ j for i ≠ j , i , j = 1 , ... , p , rank (X) = p and both Λ and X being closed under complex conjugation, find real symmetric matrices M, D and K such that M X Λ 2 + D X Λ + K X = 0. Problem OAP: Given real symmetric matrices M a , D a and K a , find (M ^ , D ^ , K ^) ∈ S E such that ‖ M ^ − M a ‖ N 2 + ‖ D ^ − D a ‖ N 2 + ‖ K ^ − K a ‖ N 2 = min (M , D , K) ∈ S E (‖ M − M a ‖ N 2 + ‖ D − D a ‖ N 2 + ‖ K − K a ‖ N 2) , where S E is the solution set of Problem IQEP and ‖ ⋅ ‖ N is a weighted Frobenius norm. The representation of the general solution to Problem IQEP and the explicit formula for the optimal approximate solution of Problem OAP are given by applying the singular value decomposition. A given numerical example shows that the updated model can be in good agreement with the modal test data. [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]

Let G n , β be the set of graphs of order n with given matching number β. Let D (G) be the diagonal matrix of the degrees of the graph G and A (G) be the adjacency matrix of the graph G. The largest eigenvalue of the nonnegative matrix A α (G) = α D (G) + A (G) is called the α-spectral radius of G. The graphs with maximal α-spectral radius in G n , β are completely characterized in this paper. In this way, we provide a general framework to attack the problem of extremal spectral radius in G n , β . More precisely, we generalize the known results on the maximal adjacency spectral radius in G n , β and the signless Laplacian spectral radius. [ABSTRACT FROM AUTHOR]

This paper studies two inverse eigenvalue problems for two kinds of acyclic matrices whose graphs are caterpillars. The spectral data of the first problem considers the minimal and maximal eigenvalues of all leading principal submatrices of the matrix. The second consists of an extremal eigenvalue of each leading principal submatrix and one eigenpair of the matrix. In the main results, we give sufficient conditions for the existence of such matrices, and their proofs provide algorithmic procedures for their construction. Finally, we present some numerical examples that illustrate the applicability of the solutions obtained. [ABSTRACT FROM AUTHOR]

Let C be a conjugation on a complex separable Hilbert space H. A bounded linear operator T is said to be C-normal if C T ∗ T C = T T ∗ . In this paper, first, we give a representation of C-normal operators on finite dimensional Hilbert space and later extend it to compact C-normal operators on infinite-dimensional separable Hilbert spaces. In the end, we discuss the eigenvalue problem for C-normal operators and show that every compact C-normal operator has a solution for the eigenvalue problem. [ABSTRACT FROM AUTHOR]

A complex unit gain graph is a triple Φ = (G , T , φ) (or G φ for short) consisting of a simple graph G, as the underlying graph of G φ , the set of unit complex numbers T = { z ∈ C : | z | = 1 } and a gain function φ : E → → T such that φ (e i , j) = φ (e j , i ) − 1 . Let A (G φ) be the adjacency matrix of G φ . In this paper, we prove that m (G) − c (G) ≤ p (G φ) ≤ m (G) + c (G) , m (G) − c (G) ≤ n (G φ) ≤ m (G) + c (G) , where p (G φ) , n (G φ) , m (G) and c (G) are the number of positive eigenvalues of A (G φ) , the number of negative eigenvalues of A (G φ) , the matching number and the cyclomatic number of G, respectively. Furthermore, we characterize the graphs which attain the upper bounds and the lower bounds, respectively. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish new log-majorization results concerning eigenvalues and singular values which generalize some previous work related to a conjecture and an open question which were presented by Lemos and Soares in 2018. In addition, we present a complement of a unitarily invariant norm inequality which was conjectured by Bhatia, Lim and Yamazaki in 2016, and recently proved by Dinh, Dumitru and Franco in 2017 for the Schatten p-norm with 1 ≤ p ≤ ∞. [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]

Two graphs having the same spectrum are said to be cospectral. A pair of singularly cospectral graphs is formed by two graphs such that the absolute values of their nonzero eigenvalues coincide. Clearly, a pair of cospectral graphs is also singularly cospectral but the converse may not be true. Two graphs are almost cospectral if their nonzero eigenvalues and their multiplicities coincide. In this paper, we present necessary and sufficient conditions for a pair of graphs to be singularly cospectral, giving an answer to a problem posted by Nikiforov. In addition, we construct an infinite family of pairs of noncospectral singularly cospectral graphs with unbounded number of vertices. It is clear that almost cospectral graphs are also singularly cospectral but the converse is not necessarily true, we present families of graphs where both concepts: almost cospectrality and singularly cospectrality agree. [ABSTRACT FROM AUTHOR]

One way to study a hypergraph is to attach to it a tensor. Tensors are a generalization of matrices, and they are an efficient way to encode information in a compact form. In this paper, we study how properties of weighted hypergraphs are reflected on eigenvalues and eigenvectors of their associated tensors. We also show how to efficiently compute eigenvalues with some techniques from numerical algebraic geometry. [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]

Let G be a simple graph on n vertices. The Laplacian Estrada index of G is defined as L E E (G) = ∑ i = 1 n e μ i (G) , where μ 1 (G) , μ 2 (G) , ... , μ n (G) are the Laplacian eigenvalues of G. In this paper, we characterize the connected (n , m) -graph with the maximum Laplacian Estrada index for m ≥ 3 n 2 − 4 n + 1 8 . We also obtain some results on the Laplacian Estrada index for 3 n − 6 ≤ m < 3 n 2 − 4 n + 1 8 . [ABSTRACT FROM AUTHOR]

In this paper, we establish a new version of a Schur–Horn type theorem. Meanwhile, we provide a new proof of a recent result in [Bhatia R, Jain T. A Schur-Horn theorem for symplectic eigenvalues. Linear Algebra Appl. 2020;599:133–139]. [ABSTRACT FROM AUTHOR]

The eccentricity matrix ϵ (G) of a connected graph G is obtained from the distance matrix of G by retaining the largest distance in each row and each column and setting the remaining entries as 0. The eccentricity matrices of graphs are closely related to the distance matrices of graphs, nevertheless a number of properties of eccentricity matrices are substantially different from those of the distance matrices. The eccentricity matrices of trees need not be invertible though the distance matrices of trees are invertible. In this paper, we establish a necessary and sufficient condition for the eccentricity matrix of a tree to be invertible. The largest eigenvalue of ϵ (G) is called the ε-spectral radius, and the eccentricity energy (or the ε-energy) of G is the sum of the absolute values of the eigenvalues of ϵ (G). We obtain some bounds for the ε-spectral radius and characterize the extreme graphs. Two graphs are said to be ε-equienergetic if they have the same ε-energy. For any n ≥ 5 , we construct a pair of ε-equienergetic graphs on n vertices, which are not ε-cospectral. [ABSTRACT FROM AUTHOR]

The classical Ostrowski Theorem gives an estimate for the eigenvalues of a matrix by a union of some disks. In this paper, we present the Ostrowski Theorem on the localization of the spectrum of an operator matrix. [ABSTRACT FROM AUTHOR]

For n ≥ 2, let S Sn* = {(0, n, 0), (0, n - 1, 1), (1, n - 1, 0), (n, 0, 0), (n - 1, 0, 1), (n - 1, 1, 0)}. A zero-nonzero pattern A of order n allows Sn* if Sn* ⊆ i(A), the inertia set of A, and requires Sn* if Sn* = i(A). The study of zero-nonzero patterns allowing Sn* was introduced by Berliner et al. [Inertia sets allowed by matrix patterns. Electron J Linear Algebra. 2018;34:343-355]. In this paper, we give characterizations of irreducible zero-nonzero path patterns and double-star patterns that allow Sn*. Moreover, all irreducible zero-nonzero tree patterns of order 5 and 6 that allow Sn* are also characterized. [ABSTRACT FROM AUTHOR]

In this paper we compute bounds for signless Laplacian energy, distance signless Laplacian eigenvalues and signless Laplacian energy of unitary addition Cayley graph Gn. We also obtain distance Laplacian eigenvalues and distance Laplacian energy of Gn. [ABSTRACT FROM AUTHOR]

Higher degree forms are homogeneous polynomials of degree d > 2, or equivalently symmetric d-linear spaces. This paper is mainly concerned about the algebraic structure of the centres of higher degree forms with applications specifically to direct sum decompositions, namely expressing higher degree forms as sums of forms in disjoint sets of variables. We show that the centre algebra of almost every form is the ground field, consequently almost all higher degree forms are absolutely indecomposable. If a higher degree form is decomposable, then we provide simple criteria and algorithms for direct sum decompositions by its centre algebra. It is shown that the direct sum decomposition problem can be boiled down to some standard tasks of linear algebra, in particular the computations of eigenvalues and eigenvectors. We also apply the structure results of centre algebras to provide a complete answer to the classical problem of whether a higher degree form can be reconstructed from its Jacobian ideal. [ABSTRACT FROM AUTHOR]

Let Fn be the fan graph with n ≥ 3 vertices. The distance d(i, j) between any two distinct vertices i and j of Fn is the length of the shortest path connecting i and j. Let ... be the n x n symmetric matrix with diagonal entries equal to zero and off-diagonal entries equal to d(i, j). In this paper, we find two positive semidefinite matrices Lo and Le such that rank(Lo) = rank(Le) = n - 1, all row sums of Lo and Le are equal to zero, and find two rank one matrices αα' and ... such that ... if n is odd; if n is even. The interlacing property between the eigenvalues of ... and Lo (resp., Le) is also established. [ABSTRACT FROM AUTHOR]

In this paper, to find the numerical solution of the linear systems with the non-Hermitian positive definite matrix, a preconditioned two-sweep shift splitting (PTSS) iteration method is introduced and its convergent conditions are discussed. When the PTSS method is used as a preconditioner, the eigenvalue distribution of the corresponding preconditioned matrix is given. Numerical experiments are reported to the efficiency of the PTSS method and the PTSS preconditioner. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph. For real number α ∈ [0, 1], Nikiforov defined the Aα-matrix of G as Aα(G) = αD(G) + (1 − α)A(G), where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G respectively. In this paper, we obtain a sharp upper bound on the largest eigenvalue ρα(G) of Aα(G) for α ∈ [1/2, 1). Employing this upper bound, we prove that 'For connected G1 and G2 with n vertices and m edges, if the maximum degree Δ(G1) ≥ 2α(1 − α)(2m − n + 1) + 2α and Δ(G1) > Δ(G2), then ρ α (G 1) > ρ α (G 2) '. Let λα(G) denote the least eigenvalue of Aα(G). For α ∈ (1/2, 1), we prove that 'For two connected G1 and G2, if the minimum degree δ (G 1) ≤ 1 1 − α − 2 and δ(G1) < δ(G2), then λα(G1) < λα(G2)'. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n. The matrix S(G) = D(G) + A(G) is called the signless Laplacian matrix of G, where D(G) and A(G) denote the diagonal matrix of vertex degrees and the adjacency matrix of G, respectively. Let s1(G) and λ1(G) be the largest eigenvalue of S(G) and A(G), respectively. In this paper, we first present sharp upper and lower bounds for s1(G) and λ1(G) involving the maximum degree, the minimum degree, order, size and sum-connectivity F-index. Moreover, we investigate the relation between s1(G) and λ1(G). [ABSTRACT FROM AUTHOR]

The symmetric Pascal matrix is a square matrix whose entries are given by binomial coefficients modulo 2. In 1997, Christopher and Kennedy defined and studied the binomial graph, which is the graph whose adjacency matrix is the symmetric Pascal matrix. They computed the spectrum of the binomial graph of order a power of 2. In this paper, we study spectral properties of the binomial graph of any order such as eigenvalues and eigenvectors, algebraic connectivities and inertia indices. We also compute the determinant of the symmetric Pascal matrix in modulo 3. [ABSTRACT FROM AUTHOR]

For a simple connected graph G of order n, let D(G), Tr(G), DL(G) and DQ(G) be, respectively, the distance matrix, the diagonal matrix of the vertex transmissions, the distance Laplacian matrix and the distance signless Laplacian matrix of G. For a real number α ∈ [0, 1], the generalized distance matrix Dα(G) is defined as Dα(G) = αTr(G) + (1 − α)D(G). In this paper, we establish relations between the smallest eigenvalues of D(G), DQ(G) and Dα(G). We obtain sharp upper and lower bounds for the smallest eigenvalue of Dα(G) in terms of various graph parameters like the order n, the diameter d, the Wiener index W(G), the chromatic number χ(G), the transmission degrees and the parameter α. We also identify the extremal graphs attaining the given bounds. [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 first study zero divisor graphs over finite chain rings. We determine their rank, determinant, and eigenvalues using reduction graphs. Moreover, we extend the work to zero divisor graphs over finite commutative principal ideal rings using a combinatorial method, finding the number of positive eigenvalues and the number of negative eigenvalues, and finding upper and lower bounds for the largest eigenvalue. Finally, we characterize all finite commutative principal ideal rings such that their zero divisor graphs are complete and compute the Wiener index of the zero divisor graphs over finite commutative principal ideal rings. [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]

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]

A graph G is called triangle-free if G does not contain a triangle as an induced subgraph. Let H n be the set of triangle-free graphs of order n with six non-zero eigenvalues. In this paper, we find 19 graphs of H n , and we show that the other graphs of H n can be constructed from these 19 graphs by adding some congruent vertices. Hence we completely characterize the triangle-free graphs with six non-zero eigenvalues. [ABSTRACT FROM AUTHOR]

The finite element model errors mainly come from the complex parts of the geometry, boundary conditions and stress state of the structure. Therefore, the problem for updating mass and stiffness matrices can be reduced to an inverse problem for symmetric matrices with submatrix constraints (IP-MUP): Let Λ = d i a g (λ 1 , ... , λ p) ∈ R p × p and Φ = [ ϕ 1 , ... , ϕ p ] ∈ R n × p be the measured eigenvalue and eigenvector matrices with rank(Φ) = p. Find n × n symmetric matrices M and K such that K Φ = M Φ Λ , Φ ⊤ M Φ = I p , s. t. M (r) = M 0 , K (r) = K 0 , where M(r) and K(r) are the r × r leading principal submatrices of M and K, respectively. We then consider an optimal approximation problem (OAP): Given n × n symmetric matrices Ma and Ka. Find (M ˆ , K ˆ) ∈ S E such that ∥ K ˆ − K a ∥ 2 + ∥ M ˆ − M a ∥ 2 = min (M , K) ∈ S E (∥ K − K a ∥ 2 + ∥ M − M a ∥ 2) , where S E is the solution set of Problem IP-MUP. In this paper, the solvability condition for Problem IP-MUP is established, and the expression of the general solution of Problem IP-MUP is derived. Also, we show that the optimal approximation solution (M ˆ , K ˆ) is unique and derive an explicit formula for it. [ABSTRACT FROM AUTHOR]