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The region in the complex plane containing the eigenvalues of all n × n stochastic matrices was described by Karpelevič in 1951, and it is since then known as the Karpelevič region. The boundary of the Karpelevič region is the union of arcs called the Karpelevič arcs. We provide a complete characterization of the Karpelevič arcs that are powers of some other Karpelevič arc. Furthermore, we find the necessary and sufficient conditions for a sparsest stochastic matrix associated with the Karpelevič arc of order n to be a power of another stochastic matrix. [ABSTRACT FROM AUTHOR]

A complex unit gain graph, or T -gain graph, is a triple Φ = (G , T , φ) comprised of a simple graph G as the underlying graph of Φ, the set of unit complex numbers T = { z ∈ C : | z | = 1 } , and a gain function φ : E → → T with the property that φ (e i j) = φ (e j i) − 1. A cactus graph is a connected graph in which any two cycles have at most one vertex in common. In this paper, we firstly show that there does not exist a complex unit gain graph with nullity n (G) − 2 m (G) + 2 c (G) − 1 , where n (G) , m (G) and c (G) are the order, matching number, and cyclomatic number of G. Next, we provide a lower bound on the nullity for connected complex unit gain graphs and an upper bound on the nullity for complex unit gain bipartite graphs. Finally, we characterize all non-singular complex unit gain bipartite cactus graphs, which generalizes a result in Wong et al. (2022) [30]. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph and A (G) be the adjacency matrix of G. A diagonal matrix with diagonal entries ±1 is called a signature matrix. If A (G) is nonsingular and X = S A (G) − 1 S − 1 is entrywise nonnegative for some signature matrix S , then X can be viewed as the adjacency matrix of a unique weighted graph. It is called the inverse of G , denoted by G +. A graph G is said to have the reciprocal eigenvalue property (property(R)) if A (G) is nonsingular, and 1 λ is an eigenvalue of A (G) whenever λ is an eigenvalue of A (G). Further, if λ and 1 λ have the same multiplicity for each eigenvalue λ , then G is said to have the strong reciprocal eigenvalue property (property (SR)). It is known that for a tree T , the following conditions are equivalent: a) T + is isomorphic to T , b) T has property (R), c) T has property (SR) and d) T is a corona tree (it is a tree which is obtained from another tree by adding a new pendant at each vertex). Studies on the inverses, property (R) and property (SR) of bipartite graphs are available in the literature. However, their studies for the non-bipartite graphs are rarely done. In this article, we study the inverse and property (SR) for non-bipartite graphs. We first introduce an operation, which helps us to study the inverses of non-bipartite graphs. As a consequence, we supply a class of non-bipartite graphs for which the inverse graph G + exists and G + is isomorphic to G. It follows that each graph G in this class has property (SR). [ABSTRACT FROM AUTHOR]

In this note, we give simple proofs of two recent extensions of the Hua determinant inequality. We also answer a question of Poon affirmatively on a Young type inequality, which appears in his investigation of inequalities on eigenvalues arising from the Hua determinant inequality. [ABSTRACT FROM AUTHOR]

There exist two major subclasses in the class of superquadratic functions, one comprises concave and decreasing functions, while the other consists of convex and monotone increasing functions. Leveraging this distinction, we introduce eigenvalue inequalities for each case. The characteristics of these functions allow us to advance our findings in two ways: firstly, by refining existing results related to eigenvalues for convex functions, and secondly, by deriving complementary inequalities for other function types. To bolster our claims, we will provide illustrative examples. [ABSTRACT FROM AUTHOR]

We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order n , where one edge has weight α ∈ C , with Re (α) < 0 , and all the others have weights 1. This paper is a sequel of a previous one where we considered Re (α) ∈ [ 0 , 1 ] (Grudsky et al., 2022 [12]). We prove that for Re (α) < 0 and n > Re (α − 1) / Re (α) , one eigenvalue is negative while the others belong to [ 0 , 4 ] and are distributed as the function x ↦ 4 sin 2 ⁡ (x / 2). Additionally, we prove that as n tends to ∞, the outlier eigenvalue converges exponentially to 4 Re (α) 2 / (2 Re (α) − 1). We give exact formulas for half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as n tends to ∞, both for the eigenvalues belonging to [ 0 , 4 ] and the outlier. We also compute the eigenvectors and their norms. [ABSTRACT FROM AUTHOR]

In this paper, we primarily focus on the eigenvalues of the adjacency matrix and Seidel matrix of chain graphs, referred to as eigenvalues and Seidel eigenvalues of these graphs, respectively. Firstly, we utilize the characteristic polynomial of the adjacency matrix of a chain graph to construct infinite pairs of non-isomorphic cospectral chain graphs. Next, we determine the inertia of the Seidel matrix of a chain graph and establish an interval that does not contain the Seidel eigenvalues of chain graphs. Lastly, we characterize chain graphs with distinct Seidel eigenvalues. [ABSTRACT FROM AUTHOR]

Let G be a finite simple graph with λ as an eigenvalue (i.e. an eigenvalue of the adjacency matrix of G), and let H be a star complement for λ in G. Motivated by a controllability condition, we say that H is a strong star complement for λ if G and H have no eigenvalue in common. We explore this concept in the context of line graphs, exceptional graphs, strongly regular graphs and graphs with a prescribed star complement. [ABSTRACT FROM AUTHOR]

This paper is concerned with establishing a trace minimization principle for two Hermitian matrix pairs. Specifically, we will answer the question: when is inf X ⁡ tr (A ˆ X H A X) subject to B ˆ X H B X = I (the identity matrix of apt size) finite? Sufficient and necessary conditions are obtained and, when the infimum is finite, an explicit formula for it is established in terms of the finite eigenvalues of the matrix pairs. Our results extend Fan's trace minimization principle (1949) for a Hermitian matrix, a minimization principle of Kovač-Striko and Veselić (1995) for a Hermitian matrix pair, and most recent ones by the authors and their collaborators for a Hermitian matrix pair and a Hermitian matrix. [ABSTRACT FROM AUTHOR]

Uniform hypergraphs have a natural one-to-one correspondence to tensors. In this paper, we investigate the Estrada index and subgraph centrality of an m -uniform hypergraph H via the adjacency tensor. We establish some bounds for the Estrada index and give expressions of the subgraph centrality in terms of graph parameters of the multi-digraphs associated with H. When H is 2-uniform, the above Estrada index and subgraph centrality are the Estrada index and subgraph centrality of a graph. [ABSTRACT FROM AUTHOR]

The distance between an eigenvalue λ of a signed graph G ̇ and the spectrum of a signed graph H ̇ is defined as min { | λ − μ | : μ is an eigenvalue of H ̇ }. In this paper, we investigate this distance when H ̇ is a largest induced subgraph of G ̇ that does not have λ as an eigenvalue. We estimate the distance in terms of eigenvectors and structural parameters related to vertex degrees. For example, we show that | λ | | λ − μ | ≤ δ G ̇ ∖ H ̇ max { d H ̇ (i) d G ̇ − V (H ̇) (j) : i ∈ V (G ̇) ∖ V (H ̇) , j ∈ V (H ̇) , i ∼ j } , where δ G ̇ ∖ H ̇ is the minimum vertex degree in V (G ̇) ∖ V (H ̇). If H ̇ is obtained by deleting a single vertex i , this bound reduces to | λ | | λ − μ | ≤ d (i). We also consider the case in which λ is a simple eigenvalue and H ̇ is not necessarily a vertex-deleted subgraph, and the case when λ is the largest eigenvalue of an ordinary (unsigned) graph. Our results for signed graphs apply to ordinary graphs. They can be interesting in the context of eigenvalue distribution, eigenvalue location or spectral distances of (signed) graphs. [ABSTRACT FROM AUTHOR]

The paper presents methods for the eigenvalue localisation of regular matrix polynomials, in particular, the stability of matrix polynomials is investigated. For this aim a stronger notion of hyperstability is introduced and widely discussed. Matrix versions of the Gauss-Lucas theorem and Szász inequality are shown. Further, tools for investigating (hyper)stability by multivariate complex analysis methods are provided. Several seconds- and third-order matrix polynomials with particular semi-definiteness assumptions on coefficients are shown to be stable. [ABSTRACT FROM AUTHOR]

The study of constraining the eigenvalues of the sum of two symmetric matrices, say P + Q , in terms of the eigenvalues of P and Q , has a long tradition. It is closely related to estimating a lower bound on the minimum singular value of a matrix, which has been discussed by a great number of authors. To our knowledge, no study has yielded a positive lower bound on the minimum eigenvalue, λ min (P + Q) , when P + Q is symmetric positive definite with P and Q singular positive semi-definite. We derive two new lower bounds on λ min (P + Q) in terms of the minimum positive eigenvalues of P and Q. The bounds take into account geometric information by utilizing the Friedrichs angles between certain subspaces. The basic result is when P and Q are two non-zero singular positive semi-definite matrices such that P + Q is non-singular, then λ min (P + Q) ≥ (1 − cos ⁡ θ F) min ⁡ { λ min (P) , λ min (Q) } , where λ min represents the minimum positive eigenvalue of the matrix, and θ F is the Friedrichs angle between the range spaces of P and Q. Such estimates lead to new lower bounds on the minimum singular value of full rank 1 × 2 , 2 × 1 , and 2 × 2 block matrices. Some examples provided in this paper further highlight the simplicity of applying the results in comparison to some existing lower bounds. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a simple graph with the vertex set V and the edge set E (| V | = n ; | E | = m). The automorphism group Aut (G) of G is represented by the permutation group P ≅ Aut (G) acting on the vertex set V. An arbitrary permutation p ∈ P has a cycle type 1 c 1 2 c 2 ⋯ n c n ( ∑ j = 1 n c j j = n) , where the superscript c j means that p contains exactly c j cycles of length j (j ∈ { 1 , 2 , ... , n }). We call a permutation (resp. automorphism) p ∈ P a covering permutation (automorphism) if each of its cycles (i 1 i 2) of length l = 2 corresponds to an edge v i 1 v i 2 of G , and each cycle (j 1 j 2 ⋯ j l) of length l ≥ 3 corresponds to a simple cycle v j 1 v j 2 ⋯ v j l of G (in which all vertices are different). Let α j be an A -eigenvalue (of the adjacency matrix A) of G ( α 1 > α 2 ≥ ⋯ ≥ α n ). We study the relationship between covering permutations of various cycle types and eigenvalues of the graph G. In particular, we show that if P has a covering involutive permutation h of cycle type 1 n − 2 c 2 2 c 2 , then G has c 2 eigenvalues α s q (q ∈ { 1 , 2 , ... , c 2 }) such that ∑ q = 1 c 2 α s q = − c 2 . If h is of type 1 n − 2 2 1 , then G necessarily has at least one eigenvalue − 1. Some similar results for ordinary and signless Laplacian spectra of G are also discussed. [ABSTRACT FROM AUTHOR]

Let T be a linear tree, and let S (T) denote the set of real symmetric matrices whose graph is T. If U (T) is the minimum number of eigenvalues with multiplicity 1 among matrices in S (T) and T ′ is a linear tree resulting from the addition of one vertex to T , we show that | U (T ′) − U (T) | ⩽ 1. We also determine the exact set of possible values of U (T) − U (T ′) , depending upon the manner in which the vertex is added to T to get T ′. These results are then used to give a new bound for U (T) , the diameter bound, and to improve an existing bound, 2 + D 2 (T). [ABSTRACT FROM AUTHOR]

A multigraph is a graph with possible multiple edges, but no loops. Let t be a positive integer. Let G t be the set of simple graphs (or multigraphs) such that for each G ∈ G t there exist at least t + 1 non-empty disjoint proper subsets V 1 , V 2 , ... , V t + 1 ⊆ V (G) satisfying V (G) ∖ (V 1 ∪ V 2 ∪ ⋯ ∪ V t + 1) ≠ ϕ and edge connectivity κ ′ (G) = e (V i , V (G) ∖ V i) for i = 1 , 2 , ... , t + 1. Let D (G) and A (G) denote the degree diagonal matrix and adjacency matrix of a simple graph (or a multigraph) G , and let μ i (G) be the i th largest eigenvalue of the Laplacian matrix L (G) = D (G) + A (G). In this paper, we investigate the relationship between μ n − 2 (G) and edge connectivity or spanning tree packing number of a graph G ∈ G 1 , respectively. We also give the relationship between μ n − 3 (G) and edge connectivity or spanning tree packing number of a graph G ∈ G 2 , respectively. Moreover, we generalize all the results about L (G) to a more general matrix a D (G) + A (G) , where a is a real number with a ≥ − 1. [ABSTRACT FROM AUTHOR]

A signed graph Σ = (G , σ) is a graph where the function σ assigns either 1 or −1 to each edge of the simple graph G. The adjacency matrix of Σ, denoted by A (Σ) , is defined canonically. In a recent paper, Wang et al. extended the spectral bounds of Hoffman and Cvetković for the chromatic number of signed graphs. They proposed an open problem related to the balanced clique number and the largest eigenvalue of a signed graph. We solve a strengthened version of this open problem. As a byproduct, we give alternate proofs for some of the known classical bounds for the least eigenvalues of unsigned graphs. We extend the Turán's inequality for the signed graphs. Besides, we study the Bollobás and Nikiforov conjecture for the signed graphs and show that the conjecture need not be true for the signed graphs. Nevertheless, the conjecture holds for signed graphs under some assumptions. Finally, we study some of the relationships between the number of signed walks and the largest eigenvalue of a signed graph. [ABSTRACT FROM AUTHOR]

The anti-adjacency (aka eccentricity) matrix A (G) of a graph G is obtained from the distance matrix by retaining the eccentricities (the largest distance) in each row and each column. The motivation for this paper is to find answers to two problems: "For which connected graphs the eccentricity (anti-adjacency) matrix is either reducible or irreducible?" and "Characterize the graphs with a small number of distinct anti-adjacency eigenvalues". In this context, we get the sufficient and necessary conditions of the graphs which have an irreducible matrix. For the second problem, we characterize the graphs with exactly two distinct eigenvalues different from 0. We also obtain some spectral properties of the anti-adjacency matrix. [ABSTRACT FROM AUTHOR]

Let H be a uniform hypergraph with cut vertices, which is a coalescence of two nontrivial connected sub-hypergraphs (called branches) at some vertex. The traces of the adjacency tensor A (H) of H , simply called the traces of H , are important in the expression of the determinant and the characteristic polynomial of A (H) , and are closely related to the Estrada index of H. In this paper we give a formula for the traces of H in terms of those of its branches, and get some perturbation results on the traces of H when a branch of H attached at one vertex is relocated to another vertex. We prove that among all hypertrees with fixed number of edges, the hyperpath is the unique one with minimum Estrada index and the hyperstar is the unique one with maximum Estrada index. [ABSTRACT FROM AUTHOR]

We classify maximal digraphs whose Hermitian spectral radius is at most 2. [ABSTRACT FROM AUTHOR]

The Generalized Simplified Spherical Harmonics (GSP N) method has drawn recent interest owing to the fact that theoretically, it is equivalent to the P N and does not rely on any assumption other than piecewise homogeneity. Its level wise approach to P N offers an agreeable balance between the accuracy and computational efficiency making it a valuable mathematical tool for analyzing neutron transport problems pertaining to nuclear engineering and reactor physics. The lowest level approximation for N = 3, expressed as GSP 3 (0) , offers reducing the complexity while still capturing the essential physics. In the current work, GSP 3 (0) equations are derived along with the interface and boundary conditions for linear anisotropic scattering. These equations are then discretized using the generalized transverse integration nodal method for a two-dimensional system of piecewise homogeneous rectangular regions. The discretization is done with an option kept for reducing the GSP 3 (0) formalism to the conventional SP 3. Subsequently, a computer code has been developed to solve these discretized equations in the multigroup structure for vacuum, reflective or albedo boundaries. At present, this switchable SP 3 / GSP 3 (0) code has the capability of estimating k -eigenvalue and position dependent scalar neutron flux within a given two-dimensional rectangular geometry. The code has been verified by solving SP 3 , GSP 3 (0) and other benchmark problems from available literature. The results obtained from our code demonstrate that GSP 3 (0) is superior to the conventional SP 3 and comparable to the recently proposed SDP N (N = 1,2,3) [M. Nazari, A. Zolfaghari, M. Abbasi, Prog. Nucl. Energy, 2023; 166 , 104,933] in terms of accuracy as well as computation time. • Derivation of GSP 3 (0) equations for linear anisotropic scattering along with interface and boundary conditions. • Discretization in 2D rectangular geometry based on generalized transverse integration nodal method. • Development of a switchable computer code to solve SP 3 / GSP 3 (0) equations. • Verification of the code with benchmark problems and discussion. • Explicit expressions are provided for code development. [ABSTRACT FROM AUTHOR]

Generalized eigenvalue proximal support vector machine (GEPSVM) has attracted widespread attention due to its simple architecture, rapid execution, and commendable performance. GEPSVM gives equal significance to all samples, thereby diminishing its robustness and efficacy when confronted with real-world datasets containing noise and outliers. In order to reduce the impact of noises and outliers, we propose a novel intuitionistic fuzzy generalized eigenvalue proximal support vector machine (IF-GEPSVM). The proposed IF-GEPSVM assigns the intuitionistic fuzzy score to each training sample based on its location and surroundings in the high-dimensional feature space by using a kernel function. The solution of the IF-GEPSVM optimization problem is obtained by solving a generalized eigenvalue problem. Further, we propose an intuitionistic fuzzy improved generalized eigenvalue proximal support vector machine (IF-IGEPSVM) by solving standard eigenvalue decomposition resulting in simpler optimization problems with less computation cost which leads to an efficient intuitionistic fuzzy-based model. We conduct a comprehensive evaluation of the proposed IF-GEPSVM and IF-IGEPSVM models on UCI and KEEL benchmark datasets. Moreover, to evaluate the robustness of the proposed IF-GEPSVM and IF-IGEPSVM models, label noise is introduced into some UCI and KEEL datasets. The experimental findings showcase the superior generalization performance of the proposed IF-GEPSVM and IF-IGEPSVM models when compared to the existing baseline models, both with and without label noise. Our experimental results, supported by rigorous statistical analyses, confirm the superior generalization abilities of the proposed IF-GEPSVM and IF-IGEPSVM models over the baseline models. Furthermore, we implement the proposed IF-GEPSVM and IF-IGEPSVM models on the USPS recognition dataset, yielding promising results that underscore the models' effectiveness in practical and real-world applications. The source code of the proposed IF-GEPSVM and IF-IGEPSVM models are available at https://github.com/mtanveer1/IF-GEPSVM. [ABSTRACT FROM AUTHOR]

• A novel SSSM constrained optimal power flow model is proposed. • The analytical representation of the SSSM sensitivities is newly derived. • A joint solution approach is developed to effectively solve the SSSMC-OPF model. • The conflicting relationship between generation cost and SSSM is revealed. This paper presents a novel small-signal stability margin (SSSM) constrained optimal power flow model for generation dispatch to minimize the generation cost while retaining adequate SSSM. The SSSM constraint is described in terms of the total active load variation between an initial operating point and the critical point, which is located on the dynamic performance boundary of small-signal stability. From the existing SSSM model, where the steady-state equation and the small-signal stability equation are taken into account, a modified SSSM model is proposed to reduce the computational requirement. The sensitivity representation of SSSM with respect to operating parameters is newly derived, which makes it possible for the SSSM and steady-state optimization problems to be jointly solved. A joint solution approach is proposed to solve the small-signal stability margin constrained optimal power flow (SSSMC-OPF) model. Simulation results show that the proposed approach can effectively minimize the generation cost subject to retaining a certain level of SSSM. For an 8-machine 24-bus system and a modified practical 68-machine 2395-bus system, the generation costs of SSSMC-OPF are increased by 5.28% and 2.73%, respectively, but the SSSMs are improved by 45% and 14.41%, respectively, compared to the optimal power flow. [ABSTRACT FROM AUTHOR]

• The mathematical model in this paper includes both DC/DC and AC/DC converters. • The optimization parameters of matrix perturbation theory comprehensively consider the small perturbation stability, damping ratios and stability margins objective function. • Matrix perturbation theory can avoid the repeated formation of the system state matrix and the repeated calculation of the characteristic solution problem. • Droop coefficient optimization is an effective method to improve the stability of the system with small perturbations. This paper proposes a perturbation analysis and optimization method for the droop coefficient of multi-terminal DC distribution systems to address their weak damping and instability. Firstly, a small disturbance model of the multi-terminal DC distribution system with droop control is established. A proposed objective function for optimizing droop coefficients in DC/DC and AC/DC converters comprehensively considers factors such as small disturbance stability, damping ratio, and stability margin. Then, the eigenvalues obtained from the droop coefficients perturbation analysis of each sub-matrix module of the system are used in the optimization iteration to prevent the system matrix from being repeatedly formed and the eigenvalues from being repeatedly solved during the optimization process. Finally, the model is built in PLECS with RT-Box for experimental verification. The results demonstrate that this method can effectively enhance the stability of DC distribution systems. [ABSTRACT FROM AUTHOR]

In 2013, Ku and Wong showed that for any partitions μ and μ ′ of a positive integer n with the same first part u and the lexicographic order μ ◃ μ ′ , the eigenvalues ξ μ and ξ μ ′ of the derangement graph Γ n have the property | ξ μ | ≤ | ξ μ ′ | , where the equality holds if and only if u = 3 and all other parts are less than 3. In this article, we obtain an analogous conclusion on the eigenvalues of the perfect matching derangement graph M 2 n of K 2 n by finding a new recurrence formula for the eigenvalues of M 2 n. [ABSTRACT FROM AUTHOR]

The notion of root polynomials of a polynomial matrix P (λ) was thoroughly studied in Dopico and Noferini (2020) [6]. In this paper, we extend such a systematic approach to general rational matrices R (λ) , possibly singular and possibly with coalescent pole/zero pairs. We discuss the related theory for rational matrices with coefficients in an arbitrary field. As a byproduct, we obtain sensible definitions of eigenvalues and eigenvectors of a rational matrix R (λ) , without any need to assume that R (λ) has full column rank or that the eigenvalue is not also a pole. Then, we specialize to the complex field and provide a practical algorithm to compute them, based on the construction of a minimal state space realization of the rational matrix R (λ) and then using the staircase algorithm on the linearized pencil to compute the null space as well as the root polynomials in a given point λ 0. If λ 0 is also a pole, then it is necessary to apply a preprocessing step that removes the pole while making it possible to recover the root vectors of the original matrix: in this case, we study both the relevant theory (over a general field) and an algorithmic implementation (over the complex field), still based on minimal state space realizations. [ABSTRACT FROM AUTHOR]

Various trace minimization principles have interplayed with numerical computations for the standard eigenvalue and generalized eigenvalue problems in general, as well as important applied eigenvalue problems including the linear response eigenvalue problem from electronic structure calculation and the symplectic eigenvalue problem of positive definite matrices that play important roles in classical Hamiltonian dynamics, quantum mechanics, and quantum information, among others. In this paper, Ky Fan's trace minimization principle is extended along the line of the Brockett cost function tr (D X H A X) in X on the Stiefel manifold, where D of an apt size is positive definite. Specifically, we investigate inf X ⁡ tr (D X H A X) subject to X H B X = I k (the k × k identity matrix) or X H B X = J k , where J k = diag (± 1). We establish conditions under which the infimum is finite and when it is finite, analytic solutions are obtained in terms of the eigenvalues and eigenvectors of the matrix pencil A − λ B , where B is possibly indefinite and possibly singular, and D is also possibly indefinite. [ABSTRACT FROM AUTHOR]

This article initiates the study of the minimum number of distinct eigenvalues allowed by a sign pattern. Non-trivial examples of sign patterns that allow matrices with only one eigenvalue include the potentially nilpotent and the spectrally arbitrary sign patterns, in contrast to the inverse eigenvalue problem for graphs. Necessary digraph cycle conditions are developed for a sign pattern to have a matrix realization with exactly one eigenvalue. Certain manipulations of sign patterns provide new n × n patterns that preserve the minimum number of eigenvalues allowed, including some Jacobian methods. The 2 × 2 and the irreducible 3 × 3 sign patterns are classified according to the minimum number of eigenvalues allowed by the pattern. [ABSTRACT FROM AUTHOR]

Recently, perfect (quantum) edge state transfer (PEST, in short) was proposed as a potential model for quantum information processing. In a previous paper (X. Cao (2021) [12]), a characterization on cubelike graphs having PEST is presented. In this paper, a general characterization of abelian Cayley graphs having PEST is established by generalizing the main idea of the prior paper. Some concrete constructions of abelian Cayley graphs having PEST are provided. Additionally, some existence results are confirmed as well. Notably, we show that for every even number n , there is an abelian group G with order n , and a subset S in G such that Cay (G , S) has PEST at time π / 2. Particularly, we show that PEST may occur in some abelian Cayley graphs which don't exhibit perfect state transfer (PST). [ABSTRACT FROM AUTHOR]

In this paper we study the eigenvalues of the laplacian matrices of the cyclic graphs with one edge of weight α and the others of weight 1. We denote by n the order of the graph and suppose that n tends to infinity. We notice that the characteristic polynomial and the eigenvalues depend only on Re (α). After that, through the rest of the paper we suppose that 0 < α < 1. It is easy to see that the eigenvalues belong to [ 0 , 4 ] and are asymptotically distributed as the function g (x) = 4 sin 2 ⁡ (x / 2) on [ 0 , π ]. We obtain a series of results about the individual behavior of the eigenvalues. First, we describe more precisely their localization in subintervals of [ 0 , 4 ]. Second, we transform the characteristic equation to a form convenient to solve by numerical methods. In particular, we prove that Newton's method converges for every n ≥ 3. Third, we derive asymptotic formulas for all eigenvalues, where the errors are uniformly bounded with respect to the number of the eigenvalue. [ABSTRACT FROM AUTHOR]

Suppose that Σ is a signed graph with n vertices and m edges. Let λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n be the eigenvalues of Σ. A signed graph is called balanced if each of its cycles contains an even number of negative edges, and unbalanced otherwise. Let ω b be the balanced clique number of Σ, which is the maximum order of a balanced complete subgraph of Σ. In this paper, we prove that λ 1 ≤ 2 ω b − 1 ω b m. This inequality extends a conjecture of ordinary graphs, which was confirmed by Nikiforov (2002) [8] , to the signed case. In addition, we completely characterize the signed graphs with − 1 ≤ λ 2 ≤ 0. [ABSTRACT FROM AUTHOR]

We conjecture a new lower bound on the algebraic connectivity of a graph that involves the number of vertices of high eccentricity in a graph. We prove that this lower bound implies a strengthening of the Laplacian Spread Conjecture. We discuss further conjectures, also strengthening the Laplacian Spread Conjecture, that include a conjecture for simple graphs and a conjecture for weighted graphs. [ABSTRACT FROM AUTHOR]

As a generalization of strongly regular graphs, van Dam and Omidi [8] introduced the concept of strongly walk-regular graphs. A graph is called strongly ℓ -walk-regular if the number of walks of length ℓ from a vertex to another vertex depends only on whether the two vertices are adjacent, not adjacent, or identical. They proved that this class of graphs falls into several subclasses including connected regular graphs with four eigenvalues, which are called genuine strongly ℓ -walk-regular. In this paper, we prove that the least eigenvalue of a connected genuine strongly 3-walk-regular graph is no more than −2 and characterize all graphs reaching this upper bound. • We focus on the problem that the existence of strongly ℓ -walk-regular graphs which are not strongly 3-walk-regular. This problem is natural and important. • We prove that the least eigenvalue of a connected genuine strongly 3-walk-regular graph is no more than -2 and characterize all graphs reaching this upper bound. • The problem of determining the graphs with least eigenvalue -2 was one of the first challenges in the theory of graph spectra, and most proof of this problem was difficult. The proof of our main result is concise and easy to understand. [ABSTRACT FROM AUTHOR]

In this article, we prove an extension criterion for a local strong solution to the 3D Navier–Stokes equations that only require control of the positive part of middle eigenvalue of strain tensor in the critical endpoint Besov space, i.e., λ 2 + ∈ L 2 (0 , T ; B ̇ ∞ , ∞ − 1). This gives a positive answer to the problem proposed by Miller [1] and improves the results by Wu [2–4]. The proof relies on the identity for enstrophy growth and L p -norm estimate of the gradient of λ 2 +. [ABSTRACT FROM AUTHOR]

For positive integers a ⩽ b , an even (resp. odd) [ a , b ] -factor of a graph G is a spanning subgraph F such that a ⩽ d F (v) ⩽ b and d F (v) is even (resp. odd) for all v ∈ V (G). For a connected graph G , the toughness t (G) of G is defined as t (G) = min ⁡ { | S | / c (G − S) } , in which the minimum is taken over all proper vertex-subsets S such that G − S is disconnected, where c (G − S) denotes the number of components of G − S. In this paper, we first consider the following problem: Assume S is a vertex cut of a connected bipartite graph G and let H be a component of G − S. If the number of edges between V (H) and S (in G) is bounded above, then we establish a sharp lower bound on adjacency spectral radius λ 1 (H) , and the corresponding extremal graphs are characterized. Based on this result, on the one hand, we establish sharp upper bounds on certain eigenvalues for a bipartite graph with given maximum degree, minimum degree and edge connectivity to ensure that the bipartite graph contains an even [ a , b ] -factor or an odd [ a , b ] -factor; On the other hand, we give sufficient spectral conditions for a bipartite graph with given maximum degree, minimum degree and edge connectivity to guarantee that its toughness is 1, which improves some known ones of Cioabǎ and Wong [10] , Cioabǎ and Gu [9] , Duan and Wang [11]. [ABSTRACT FROM AUTHOR]

Let C m be a cycle with length m. The k -uniform hypercycle with length m is obtained by adding k − 2 new vertices in every edge of C m , denoted by C m , k. In this paper, we obtain some trace formulas of uniform hypercycles with length four. Moreover, we give the characteristic polynomials of uniform hypercycles with length four. • By the Matrix-Tree Theorem and BEST Theorem, etc., we present some trace formulas of uniform hypercycles with length four. • By applying the trace formulas, we study the characteristic polynomials of uniform hypercycles with length four. • We study the traces and obtain an explicit characteristic polynomial expression of uniform hypercycles with length four. [ABSTRACT FROM AUTHOR]

Weight-equitable partitions of graphs, which are a natural extension of the well-known equitable partitions, have been shown to be a powerful tool to weaken the regularity assumption in several classic eigenvalue bounds. In this work we aim to further our algebraic and computational understanding of weight-equitable partitions. We do so by showing several spectral properties and algebraic characterizations, and by providing a method to find coarse weight-equitable partitions. [ABSTRACT FROM AUTHOR]

The tropical semiring R ∪ { − ∞ } is the semiring with addition "max" and multiplication "+". Tropical quadratic forms are represented by tropical symmetric matrices. Tropical quadratic forms with three variables define tropical projective quadratic plane curves. In this paper, we characterize tropical projective quadratic plane curves in terms of the eigenvalue problem for tropical matrices. In particular, we focus on the curved part of a tropical projective quadratic plane curve, that is, the cell never contained in any tropical projective line. We first prove that algebraic eigenvalues, i.e., roots of the characteristic polynomial, of a matrix express the minimum distance from the origin to the curved part. We then show that an algebraic eigenvector with respect to the minimum algebraic eigenvalue of a matrix indicates the direction to the nearest point in the curved part from the origin. [ABSTRACT FROM AUTHOR]

A celebrated result of Karpelevič describes Θ n , the collection of all eigenvalues arising from the stochastic matrices of order n. The boundary of Θ n consists of roots of certain one-parameter families of polynomials, and those polynomials are naturally associated with the so-called reduced Ito polynomials of Types 0, I, II and III. In this paper we explicitly characterise all n × n stochastic matrices whose characteristic polynomials are of Type 0 or Type I, and all sparsest stochastic matrices of order n whose characteristic polynomials are of Type II or Type III. The results provide insights into the structure of stochastic matrices having extreme eigenvalues. [ABSTRACT FROM AUTHOR]

The spectrum of an arbitrary self-adjoint extension of the minimal linear relation associated with the discrete symplectic system in the limit point case is completely characterized by using the limiting Weyl–Titchmarsh M + (λ) -function. Furthermore, a dependence of the spectrum on a boundary condition is investigated and, consequently, several results of the singular Sturmian theory are derived. [ABSTRACT FROM AUTHOR]

Let D 2 n = 〈 a , b ∣ a n = b 2 = 1 , b a b = a − 1 〉 = { 1 , a , ... , a n − 1 , b , b a , ... , b a n − 1 } be the dihedral group of order 2 n and let S be a symmetric subset of D 2 n. The dihedral Cayley graph C a y (D 2 n , S) is the graph with vertex set V C a y (D 2 n , S) = D 2 n and edge set E C a y (D 2 n , S) = { g , h } ∣ g h − 1 ∈ S , g , h ∈ D 2 n . In this paper, the complete information for the eigenvalues and their corresponding orthonormal eigenvectors of C a y (D 2 n , S) is provided. The closed-form formulae of the Kirchhoff index and the resistance distances between any two vertices of C a y (D 2 n , S) are then derived. [ABSTRACT FROM AUTHOR]

The multimodal non-identical generators having close frequency modes may complicate the design of subsynchronous damping controller. In this paper, the Band-Pass Filter (BPF) based Subsynchronous Damping Controller (SSDC) is designed with SSSC to damp multimode SSR. The system is adapted from IEEE First Benchmark Model (FBM) with two non-identical generators in parallel. One of the generators has a six-mass mechanical system and other has a four-mass mechanical system. The SSDC using reduced number of BPFs is proposed to enhance the damping of multimodal SSR while maintaining the frequency selectivity of filters. The proposed BPF based SSDC is designed to modulate the SSSC injected voltage so as to suppress the multimode components in the line. The DQ model is developed in MATLAB-Simulink using the differential and algebraic equations of the combined system. The analysis of SSR are carried out through damping torque, eigenvalue and transient simulation. The participation factor analysis is performed to identify the torsional modes of the generators. The eigenvalue analysis is used to select the number of BPFs. The BPFs with appropriate frequency selectivity are chosen to improve the damping of all critical torsional modes in the entire range of practical compensation level. The results show that, the peak negative damping at both generators is considerably reduced with the proposed SSDC in the entire range of torsional mode frequencies through practical compensation levels. • Modeling of power system with multimachines in DQ reference frame. • Different methods to analyze SSR. • Design of robust Subsynchronous Damping Controller (SSDC) for non-identical machines with reduced number of Band Pass Filters (BPF). • Validation of effectiveness of the robust BPF based SSDC. • SSR characteristics with hybrid compensation incorporating SSSC and BPF based SSDC. [ABSTRACT FROM AUTHOR]

Given a finite group G , the character graph , denoted by Δ (G) , for its irreducible character degrees is a graph with vertex set ρ (G) which is the set of prime numbers that divide the irreducible character degrees of G , and with { p , q } being an edge if there exists a non-linear χ ∈ Irr (G) whose degree is divisible by p q. In this paper, on one hand, we proceed by discussing the graphical shape of Δ (G) when it has cut vertices or small number of eigenvalues, and on the other hand we give some results on the group structure of G with such Δ (G). Recently, Lewis and Meng proved the character graph of each solvable group has at most one cut vertex. Now, we determine the structure of character graphs of solvable groups with a cut vertex and diameter 3. Furthermore, we study solvable groups whose character graphs have at most two distinct eigenvalues. Moreover, we investigate the solvable groups whose character graphs are regular with three distinct eigenvalues. In addition, we give some lower bounds for the number of edges of Δ (G). [ABSTRACT FROM AUTHOR]

In this note we resolve three conjectures from Dehmer et al. (2019) on the comparison of distance measures based on the graph spectra, by constructing families of counterexamples and using computer search. In addition, we prove other results and conditions under which the conjectures hold true. [ABSTRACT FROM AUTHOR]

For a signed graph Σ, let χ (Σ) , λ 1 and λ n be the chromatic number, the maximum eigenvalue and the minimum eigenvalue of Σ, respectively. This paper proves that, for any nonempty signed graph Σ with n vertices, χ (Σ) ≥ max ⁡ { 1 + λ 1 | λ n | , n n − λ 1 }. These two bounds extend the classical spectral lower bounds of Hoffman and Cvetković for an ordinary graph, respectively. [ABSTRACT FROM AUTHOR]

In this paper a class of tridiagonal matrices of generalized Sylvester type are studied. We determine their eigenvalues and eigenvectors using the representation theory of sl (2 , F). [ABSTRACT FROM AUTHOR]

Hoffman's ratio bound is an upper bound for the independence number of a regular graph in terms of the eigenvalues of the adjacency matrix. The bound has proved to be very useful and has been applied many times. Hoffman did not publish his result, and for a great number of users the emergence of Hoffman's bound is a black hole. With his note I hope to clarify the history of this bound and some of its generalizations. [ABSTRACT FROM AUTHOR]