Characteristic Polynomial Research Articles

Characterization of k-spectrally monomorphic two-graphs

In this paper, we give a characterization of regular two-graphs of order n in terms of spectral data of two-graphs of order n−1 and n−2. Moreover, we prove that a two-graph of order n is regular if and only if all its induced sub-two-graphs with n−2 vertices have the same characteristic polynomial.

On level energy and level characteristic polynomial of rooted trees

On level energy and level characteristic polynomial of rooted trees

Spectral Properties of Power Graph of Dihedral Groups

This paper focuses on the power graph of dihedral groups of order $2n$, $D_{2n}$, where $n \geq 3$. We show the characteristic polynomial of the power graph corresponding to the adjacency, Laplacian, signless Laplacian, and normalized form of these matrices.

Characteristic Polynomial of Power Graph for Dihedral Groups Using Degree-Based Matrices

A fundamental feature of spectral graph theory is the correspondence between matrix and graph. As a result of this relation, the characteristic polynomial of the graph can be formulated. This research focuses on the power graph of dihedral groups using degree-based matrices. Throughout this paper, we formulate the characteristic polynomial of the power graph of dihedral groups based on seven types of graph matrices which include the maximum degree, the minimum degree, the greatest common divisor degree, the first Zagreb, the second Zagreb, the misbalance degree, and the Nirmala matrices.

Minimum Degree Energy of Graphs and Pebbling Graphs

Objectives: The minimum degree energy concept is applied to pebbling graphs. This study establishes a connection between the minimum degree energy of graphs and the minimum degree energy of pebbling graphs by applying the lowest degree energy of pebbling notion to twenty standard graphs. Methods: The minimum degree energy of pebbling graph with the matrix whose was calculated using . The characteristic polynomial of the minimum degree, is must be found from the matrix. Next, the eigen values of the matrix were calculated using and the sum of all the eigen values gives the minimum degree energy. The lower and upper bounds for the minimum degree energy of graphs are established along with the algorithm for computing the minimum degree energy of graphs. Findings: The lower and upper bounds were found for the minimum degree energy of pebbling graphs. For twenty standard graphs and pebbling graphs, the minimum degree energy values were calculated, and their relation was tabulated. Novelty: Pebbling graphs were subjected to the minimal energy idea and a relationship was found between the minimum energy of pebbling graphs and the minimum energy of graphs in general. Keywords: Energy, Minimum degree energy, Grotzsch graph, Pebbling graph, Data mining

ON MATRICES ARISING IN FINITE FIELD HYPERGEOMETRIC FUNCTIONS

Abstract Lehmer [‘On certain character matrices’, Pacific J. Math.6 (1956), 491–499, and ‘Power character matrices’, Pacific J. Math.10 (1960), 895–907] defines four classes of matrices constructed from roots of unity for which the characteristic polynomials and the kth powers can be determined explicitly. We study a class of matrices which arise naturally in transformation formulae of finite field hypergeometric functions and whose entries are roots of unity and zeroes. We determine the characteristic polynomial, eigenvalues, eigenvectors and kth powers of these matrices. The eigenvalues are natural families of products of Jacobi sums.

Nonvanishing of second coefficients of Hecke polynomials

Let Tm(N,2k) be the mth Hecke operator on the space S(N,2k) of cuspforms of weight 2k and level N. This paper shows that in all but finitely many cases, which we list, the second coefficient of the characteristic polynomial of T2(N,2k) does not vanish when 2 and N are coprime.

Averages of products of characteristic polynomials and the law of real eigenvalues for the real Ginibre ensemble

Averages of products of characteristic polynomials and the law of real eigenvalues for the real Ginibre ensemble

Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices

Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices

Reciprocal eigenvalue properties using the zeta and Möbius functions

In this paper, we develop a new approach to study the spectral properties of Boolean graphs using the zeta and Möbius functions on the Boolean algebra Bn of order 2n. This approach yields new proofs of the previously known results about the reciprocal eigenvalue property of Boolean graphs. Further, this approach allows us to extend the results to a more general setting of the zero-divisor graphs Γ(P) of complement-closed and convex subposets P of Bn. To do this, we consider the left linear representation of the incidence algebra of a poset P on the vector space of all real-valued functions V(P) on P. We then write down the adjacency operator A of the graph Γ(P) as the composition of two linear operators on V(P), namely, the operator that multiplies elements of V(P) on the left by the zeta function ζ of P and the complementation operator. This allows us to obtain the determinant of A and the inverse of A in terms of the Möbius function μ of the complement-closed posets P. Additionally, if we impose convexity on the poset P, then we obtain the strong reciprocal or strong anti-reciprocal eigenvalue property of Γ(P) and also obtain the absolute palindromicity of the characteristic polynomial of A. This produces a large family of examples of graphs having the strong reciprocal or strong anti-reciprocal eigenvalue property.

Secure Multi-Party Linear Algebra with Perfect Correctness

We present new secure multi-party computation protocols for linear algebra over a finite field, which improve the state-of-the-art in terms of security. We look at the case of unconditional security with perfect correctness, i.e., information-theoretic security without errors. We notably propose an expected constant-round protocol for solving systems of m linear equations in n variables over Fq with expected complexity O(k n^2.5 + k m) (where complexity is measured in terms of the number of secure multiplications required) with k > m(m+n)+1. The previous proposals were not error-free: known protocols can indeed fail and thus reveal information with probability Omega(poly(m)/q). Our protocols are simple and rely on existing computer-algebra techniques, notably the Preparata-Sarwate algorithm, a simple but poorly known “baby-step giant-step” method for computing the characteristic polynomial of a matrix, and techniques due to Mulmuley for error-free linear algebra in positive characteristic.

Joint Moments of Higher Order Derivatives of CUE Characteristic Polynomials I: Asymptotic Formulae

Abstract We derive explicit asymptotic formulae for the joint moments of the $n_{1}$-th and $n_{2}$-th derivatives of the characteristic polynomials of Circular Unitary Ensemble random matrices for any non-negative integers $n_{1}, n_{2}$. These formulae are expressed in terms of determinants whose entries involve modified Bessel functions of the first kind. We also express them in terms of two types of combinatorial sums. Similar results are obtained for the analogue of Hardy’s $Z$-function. We use these formulae to formulate general conjectures for the joint moments of the $n_{1}$-th and $n_{2}$-th derivatives of the Riemann zeta-function and of Hardy’s $Z$-function. Our conjectures are supported by comparison with results obtained previously in the number theory literature.

Laplacian coefficients, Kirchhoff index and the number of spanning trees of graphs

Let [Formula: see text], [Formula: see text], be a simple connected graph of order [Formula: see text] and size [Formula: see text]. Denote with [Formula: see text] eigenvalues of the Laplacian matrix [Formula: see text] of [Formula: see text]. The Kirchhoff index and the number of spanning trees of [Formula: see text] expressed in terms of Laplacian eigenvalues are given by [Formula: see text] and [Formula: see text], respectively. The characteristic polynomial of [Formula: see text] is given by [Formula: see text]. The first five Laplacian coefficients have been computed in the literature. In this study, we compute the sixth Laplacian coefficient of [Formula: see text]. Then, we use it to improve the previously obtained results on [Formula: see text] and [Formula: see text]. In addition, we present new Nordhaus–Gaddum-type inequalities for [Formula: see text] and [Formula: see text].

Rogue wave pattern of multi-component derivative nonlinear Schrödinger equations.

This paper studies the multi-component derivative nonlinear Schrödinger (n-DNLS) equations featuring nonzero boundary conditions. Employing the Darboux transformation method, we derive higher-order vector rogue wave solutions for the n-DNLS equations. Specifically, we focus on the distinctive scenario where the (n+1)-order characteristic polynomial possesses an explicit (n+1)-multiple root. Additionally, we provide an in-depth analysis of the asymptotic dynamic behaviors and pattern classification inherent to the higher-order vector rogue wave solution of the n-DNLS equations, mainly when one of the internal arbitrary parameters is extremely large. These patterns are related to the root structures in the generalized Wronskian-Hermite polynomial hierarchies.

New Insights into Aromaticity through Novel Delta Polynomials and Delta Aromatic Indices

We have developed novel polynomials called delta polynomials, which are, in turn, derived from the characteristic and matching polynomials of graphs associated with polycyclic aromatic compounds. Natural logarithmic aromatic indices are derived from these delta polynomials, which are shown to provide new insights into the aromaticity of polycyclic aromatic compounds, including the highly symmetric C60 buckminsterfullerene, several other fullerenes, graphene, kekulene series and other cycloarenes, such as polycyclic circumcoronaphenes and coronoids. The newly developed aromatic index yields a value of 6.77 for graphene, 6.516865 for buckminsterfullerene C60(Ih), 5.914023 for kekulene (D6h symmetry), 6.064420 for coronene (D6h), 6.137828 for circumcoronene (D6h), 6.069668 for dicronylene and so forth. Hence, the novel scaled logarithmic aromatic delta indices developed here appear to provide good quantitative measures of aromaticity, especially when they are used in conjunction with other aromatic indicators.

Generalized Inversion of Nonlinear Operators

AbstractInversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most notable is the Moore–Penrose inverse, widely used in physics, statistics, and various fields of engineering. This work investigates generalized inversion of nonlinear operators. We first address broadly the desired properties of generalized inverses, guided by the Moore–Penrose axioms. We define the notion for general sets and then a refinement, termed pseudo-inverse, for normed spaces. We present conditions for existence and uniqueness of a pseudo-inverse and establish theoretical results investigating its properties, such as continuity, its value for operator compositions and projection operators, and others. Analytic expressions are given for the pseudo-inverse of some well-known, non-invertible, nonlinear operators, such as hard- or soft-thresholding and ReLU. We analyze a neural layer and discuss relations to wavelet thresholding. Next, the Drazin inverse, and a relaxation, are investigated for operators with equal domain and range. We present scenarios where inversion is expressible as a linear combination of forward applications of the operator. Such scenarios arise for classes of nonlinear operators with vanishing polynomials, similar to the minimal or characteristic polynomials for matrices. Inversion using forward applications may facilitate the development of new efficient algorithms for approximating generalized inversion of complex nonlinear operators.

Deletion-contraction for a unified Laplacian and applications

We define a graph Laplacian with vertex weights in addition to the more classical edge weights, which unifies the combinatorial Laplacian and the normalised Laplacian. Moreover, we give a combinatorial interpretation for the coefficients of the weighted Laplacian characteristic polynomial in terms of weighted spanning forests and use this to prove a deletion-contraction relation. We prove various interlacing theorems relating to deletion and contraction, as well as to rectangular tilings, drawing on the work of Brooks, Smith, Stone and Tutte on square tilings. Additionally, we show that the weighted Laplacian also satisfies a vertex analogue of deletion-contraction. We give applications of weighted Laplacian eigenvalues to sparse cuts, independent sets and graph colouring, and establish new cases of a problem of Stanley on distinguishing nonisomorphic trees.

Moments of zeta and correlations of divisor‐sums: Stratification and Vandermonde integrals

AbstractWe refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta‐function. This expression is analogous to a formula, recently discovered by Brad Rodgers and Kannan Soundararajan, for moments of characteristic polynomials of random matrices from the unitary group.

On the Structure of Laplacian Characteristic Polynomial of Circulant Graphs

On the Structure of Laplacian Characteristic Polynomial of Circulant Graphs

A Strong Gram Classification of Non-negative Unit Forms of Dynkin Type 𝔸r

An integral quadratic form q : ℤn → ℤ is usually identified with a bilinear form 𝕓∨q:ℤn×ℤn→ℤ satisfying q(x)=𝕓∨q(x,x) for any vector x in ℤn, and such that its Gram matrix with respect to the canonical basis of ℤn is upper triangular. Two integral quadratic forms are called strongly (resp. weakly) Gram congruent if their corresponding upper triangular bilinear forms (resp. their symmetrizations) are equivalent. If q is unitary, such upper triangular bilinear form is unimodular, and one considers the associated Coxeter transformation and its characteristic polynomial, the so-called Coxeter polynomial of q with this identification. Two strongly Gram congruent quadratic unit forms are weakly Gram congruent and have the same Coxeter polynomial. Here we show that the converse of this statement holds for the connected non-negative case of Dynkin type 𝔸r (r ≥ 1) and arbitrary corank, and use this characterization to complete a combinatorial classification of such quadratic forms started in [Fundamenta Informaticae 184(1):49–82, 2021] and [Fundamenta Informaticae 185(3):221–246, 2022].