Search Results (70)

A graph is planar if it can be embedded in the plane such that its edges intersect only at their common endpoints. In this paper, we determine the graphs attaining the second and third largest signless Laplacian spectral radii among all planar graphs of order $n\ge 398$. Furthermore, we apply this characterization to identify the planar graphs that achieve the first three largest values of the sum of the first and second largest signless Laplacian eigenvalues. Full article

Let $\mathcal{Z}$ be a graph of order n with m edges. Let $A_{ag}\left(\mathcal{Z}\right)$ represents the arithmetic–geometric matrix of $\mathcal{Z}$. The eigenvalues of the matrix $A_{ag}\left(\mathcal{Z}\right)$ are called the arithmetic–geometric eigenvalues, and the eigenvalue with the largest modulus is called the arithmetic–geometric spectral radius of $\mathcal{Z}$. The sum of the absolute values of the arithmetic–geometric eigenvalues is called the arithmetic–geometric energy of $\mathcal{Z}$. In this paper, we establish sharp upper and lower bounds for the AM-GM spectral radius in terms of various graph parameters and provide a complete characterization of the extremal graphs that attain these bounds. Additionally, we derive new bounds for the AM-GM energy of a graph and identify the corresponding extremal structures. In both contexts, our results significantly improve upon several existing bounds reported in the literature. Full article

The vibration monitoring of scaled models in wind tunnels is critical for aerodynamic testing and structural safety. The abrupt onset of flutter or other aeroelastic instabilities poses a significant risk, necessitating the development of real-time, model-agnostic monitoring systems. This paper proposes a novel, generalized health indicator (HI) based on an improved Fisher Discriminant Analysis (FDA) framework for vibration state classification. The core innovation lies in reformulating the FDA objective function to distinguish between stable and dangerous vibration states, rather than tracking degradation trends. To ensure cross-model applicability, a frequency-wise standardization technique is introduced, normalizing spectral amplitudes based on the statistics of a model’s stable state. Furthermore, a dual-mode entropic regularization term is incorporated into the optimization process. This term balances the dispersion of weights across frequency bands (promoting generalizability and avoiding overfitting to specific frequencies) with the concentration of weights on the most informative resonance frequencies (enhancing the sensitivity to dangerous states). The optimal frequency weights are obtained by solving a regularized generalized eigenvalue problem, and the resulting HI is the weighted sum of the standardized frequency amplitudes. The method is validated using simulated spectral data and flight data from a wind tunnel test, demonstrating a superior performance in the early detection of dangerous vibrations and the clear interpretability of critical frequency bands. Comparisons with traditional sparse measures and machine-learning methods highlight the proposed method’s advantages in trendability, robustness, and unique capability for cross-model adaptation. Full article

Let $A_{ex}\left(\mathcal{G}\right)$ be the extended adjacency matrix of G. The eigenvalues of $A_{ex}\left(\mathcal{G}\right)$ are called extended adjacency eigenvalues of $\mathcal{G}$. The sum of the absolute values of eigenvalues of the $A_{ex}$-matrix is called the extended adjacency energy ${\mathcal{E}}_{ex}\left(\mathcal{G}\right)$ of $\mathcal{G}$. In this paper, we obtain the $A_{ex}$-spectrum of the joined union of regular graphs in terms of their adjacency spectrum and the eigenvalues of an auxiliary matrix. Consequently, we derive the $A_{ex}$-spectrum of the join of two regular graphs, the lexicographic product of regular graphs, and the $A_{ex}$-spectrum of various families of graphs. Further, as applications of our results, we construct infinite classes of infinite families of extended adjacency equienergetic graphs. We show that the $A_{ex}$-energy of the join of two regular graphs is greater than or equal to their energy. We also determine the $A_{ex}$-eigenvalues of the power graph of finite abelian groups. Full article

Cayley graphs sit at the intersection of algebra, geometry, and theoretical computer science. Their spectra encode fine structural information about both the underlying group and the graph itself. Building on classical work of Alon–Milman, Dodziuk, Margulis, Lubotzky–Phillips–Sarnak, and many others, we develop a unified representation-theoretic framework that yields several new results. We establish a monotonicity principle showing that the algebraic connectivity never decreases when generators are added. We provide closed-form spectra for canonical 3-regular dihedral Cayley graphs, with exact spectral gaps. We prove a quantitative obstruction demonstrating that bounded-degree Cayley graphs of groups with growing abelian quotients cannot form expander families. In addition, we present two universal comparison theorems: one for quotients and one for direct products of groups. We also derive explicit eigenvalue formulas for class-sum-generating sets together with a Hoffman-type second-moment bound for all Cayley graphs. We also establish an exact relation between the Laplacian spectra of a Cayley graph and its complement, giving a closed-form expression for the complementary spectral gap. These results give new tools for deciding when a given family of Cayley graphs can or cannot expand, sharpening and extending several classical criteria. Full article

The first Zagreb index of a graph G is defined as the sum of the squares of the degrees of all the vertices in G. The Laplacian spectral radius of a graph G is defined as the largest eigenvalue of the Laplacian matrix of the graph G. In this paper, we first establish inequalities on the first Zagreb index and the Laplacian spectral radius of a graph. Using the ideas of proving the inequalities, we present sufficient conditions involving the first Zagreb index and the Laplacian spectral radius for some Hamiltonian properties of graphs. Full article

The growing importance of social networks has led to increased research into trust estimation and interpretation among network entities. It is important to predict the trust score between users in order to minimize the risks in user interactions. This article enables the identification of the most reliable and least reliable entities in a network by expressing trust scores numerically. In this paper, the social network is modeled as a graph, and trust scores are calculated by taking the powers of the ratio matrix between entities and summing them. Taking the power of the proportion matrix based on the number of entities in the network requires a lot of arithmetic load. After taking the powers of the eigenvalues of the ratio matrix, these are multiplied by the eigenvector matrix to obtain the power of the ratio matrix. In this way, the arithmetic cost required for calculating trust between entities is reduced. This paper calculates the trust score between entities using linear algebra techniques to reduce the arithmetic load. Trust detection algorithms use shortest paths and similar methods to eliminate paths that are deemed unimportant, which makes the result questionable because of the loss of data. The novelty of this method is that it calculates the trust score without the need for explicit path numbering and without any data loss. Full article

This paper presents a method for optimizing the inertia constants and damping coefficients of interconnected virtual synchronous generators (VSGs) using a genetic algorithm. The goal of optimization is to find a balance between minimizing the rate of change of frequency (RoCoF) and enhancing controllability. Five controllability-based metrics are tested: the minimum eigenvalue, the sum of the two smallest eigenvalues, the maximum eigenvalue, the trace, and the determinant of the controllability Gramian matrix. The approach includes the oscillatory modes’ damping ratio constraints to ensure the small-signal stability of the entire system. The results of optimization on the IEEE 9-bus system with three VSGs show that the proposed method improves controllability, reduces RoCoF, and maintains the desired oscillation damping. The proposed approach was tested through time-domain simulations. Full article

Studies of subradiance in a chain N two-level atoms in the single excitation regime focused mainly on the complex spectrum of the effective Hamiltonian, identifying subradiant eigenvalues. This can be achieved by finding the eigenvalues N of the Hamiltonian or by evaluating the expectation value of the Hamiltonian on a generalized Dicke state, depending on a continuous variable k. This has the advantage that the sum above N can be calculated exactly, such that N becomes a simple parameter of the system and no longer the size of the Hilbert space. However, the question remains how subradiance emerges from atoms initially excited or driven by a laser. Here we study the dynamics of the system, solving the coupled-dipole equations for N atoms and evaluating the probability to be in a generalized Dicke state at a given time. Once the subradiant regions have been identified, it is simple to see if subradiance is being generated. We discuss different initial excitation conditions that lead to subradiance and the case of atoms excited by switching on and off a weak laser. This may be relevant for future experiments aimed at detecting subradiance in ordered systems. Full article

This paper explores the common neighborhood energy ($E_{CN}\left(\mathsf{\Gamma }\right)$) of graphs derived from the dihedral group $D_{2n}$ and generalized quaternion group $Q_{4n}$, specifically the non-commuting graph (NCM-graph) and the commuting graph (CM-graph). Studying graphs associated with groups offers a powerful approach to translating algebraic properties into combinatorial structures, enabling the application of graph-theoretic tools to understand group behavior. The common neighborhood energy, defined as the sum of the absolute values of the eigenvalues of the common neighborhood (CN) matrix, i.e., ${\sum }_{i=1}^p\left|{\zeta }_i\right|$, where ${\left\{{\zeta }_i\right\}}_{i=1}^p$ are the CN eigenvalues, provides insights into the structural properties of these graphs. We derive explicit formulas for the CN characteristic polynomials and corresponding CN eigenvalues for both the NCM-graph and CM-graph as functions of n. Consequently, we establish closed-form expressions for the $E_{CN}$ of these graphs, which are parameterized by n. The validity of our theoretical results is confirmed through computational examples. This study contributes to the spectral analysis of algebraic graphs, demonstrating a direct connection between the group-theoretic structure of $D_{2n}$ and $Q_{4n}$, as well as the combinatorial energy of their associated graphs, thus furthering the understanding of group properties through spectral graph theory. Full article

Tempered fractional diffusion equations constitute a critical class of partial differential equations with broad applications across multiple physical domains. In this paper, the Crank–Nicolson method and the tempered weighted and shifted Grünwald formula are used to discretize the tempered fractional diffusion equations. The discretized system has the structure of the sum of the identity matrix and a diagonal matrix multiplied by a symmetric positive definite (SPD) Toeplitz matrix. For the discretized system, we propose a structure approximation-based preconditioning method. The structure approximation lies in two aspects: the inverse approximation based on the row-by-row strategy and the SPD Toeplitz approximation by the $\tau$ matrix. The proposed preconditioning method can be efficiently implemented using the discrete sine transform (DST). In spectral analysis, it is found that the eigenvalues of the preconditioned coefficient matrix are clustered around 1, ensuring fast convergence of Krylov subspace methods with the new preconditioner. Numerical experiments demonstrate the effectiveness of the proposed preconditioner. Full article

In this paper, theories, algorithms and properties of eigenvalues of quaternion tensors via the t-product termed T-eigenvalues are explored. Firstly, we define the T-eigenvalue of quaternion tensors and provide an algorithm to compute the right T-eigenvalues and the corresponding T-eigentensors, along with an example to illustrate the efficiency of our algorithm by comparing it with other methods. We then study some inequalities related to the right T-eigenvalues of Hermitian quaternion tensors, providing upper and lower bounds for the right T-eigenvalues of the sum of a pair of Hermitian tensors. We further generalize the Weyl theorem from matrices to quaternion third-order tensors. Additionally, we explore estimations related to right T-eigenvalues, extending the Geršgorin theorem for matrices to quaternion third-order tensors. Full article

This study explores the degree energy of fuzzy graphs to establish fundamental spectral bounds and characterize adjacency structures. We derive upper bounds on the sum of squared degree eigenvalues based on vertex degree distributions and formulate constraints using the characteristic polynomial of the maximum degree matrix. Furthermore, we demonstrate that the average degree energy of a connected fuzzy graph remains strictly positive. The proposed framework is applied to protein–protein interaction networks, identifying critical proteins and enhancing network resilience analysis. Full article

This study concerns Riemannian manifolds with two types of tangent vectors. Let it be given that there are two subspaces of a tangent space with the property that each tangent vector has a unique decomposition as the sum of a vector in one subspace and a vector in the other subspace. Then, these tangent spaces can be complexified in such a way that the theory of the modular operator applies and that the complexified subspaces are invariant for the modular automorphism group. Notions coming from Kubo–Mori theory are introduced. In particular, the admittance function and the inner product of the Kubo–Mori theory can be generalized to the present context. The parallel transport operators are complexified as well. Suitable basis vectors are introduced. The real and imaginary contributions to the connection coefficients are identified. A version of the fluctuation–dissipation theorem links the admittance function to the path dependence of the eigenvalues and eigenvectors of the Hamiltonian generator of the modular automorphism group. Full article

The concept of the energy of a graph has been widely explored in the field of mathematical chemistry and is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of a hypergraph is the trace norm of its connectivity matrices, which generalize the concept of graph energy. In this paper, we establish bounds for the adjacency energy of hypergraphs in terms of the number of vertices, maximum degree, eigenvalues, and the norm of the adjacency matrix. Additionally, we compute the sum of squares of adjacency eigenvalues of linear k-hypergraphs and derive its bounds for k-hypergraph in terms of number of vertices and uniformity of the k-hypergraph. Moreover, we determine the Nordhaus–Gaddum type bounds for the adjacency energy of k-hypergraphs. Full article