Integer Eigenvalues Research Articles

Self-Invertible Key on Cipher Polygraphic Polyfunction with Eigen Matrix based IMIE

Hill Cipher’s System and its modifications are still practiced mainly in sending a secret message involving images. One drawback that the recipient of the message is trying to overcome is to get the decryption key from the cipher text to plain text during the decryption process. Previous studies have proven that using self-invertible keys can reduce the computation cost to obtain this key. This paper generates a self-invertible matrix based on Integer-Entry Matrix with all Integer Eigenvalues (IMIE). Subsequently, we executed it into Cipher Polygraphic Polyfunction Cryptosystem and observed the effect.

Partially hyperbolic endomorphisms with expanding linear part

Abstract In this paper, we study transitivity of partially hyperbolic endomorphisms of the two torus whose action in the first homology group has two integer eigenvalues of moduli greater than one. We prove that if the Jacobian is everywhere greater than the modulus of the largest eigenvalue, then the map is robustly transitive. For this, we introduce Blichfedt’s theorem as a tool for extracting dynamical information from the action of a map in homology. We also treat the case of specially partially hyperbolic endomorphisms, for which we obtain a complete dichotomy: either the map is transitive and conjugated to its linear part, or its unstable foliation must contain an annulus which may either be wandering or periodic.

On integer matrices with integer eigenvalues and Laplacian integral graphs

A matrix A is said to be an integer matrix if all its entries are integers. In this article, we characterize the nonsingular integer matrices with integer eigenvalues in an expressible form of their corresponding inverse matrices. It is proved that a nonsingular integer matrix A has integer eigenvalues if and only if A−1 can be written as the sum of n rank-one matrices that meet certain requirements. A method for constructing integer matrices with integer eigenvalues using the Hadamard product is also provided. Let S represent an n-tuple of nonnegative integers. If there is an n×n integer matrix A whose spectrum (the collection of eigenvalues) is S, we say that S is realisable by an integer matrix. In Fallat et al. (2005) [7], the authors posed a conjecture that “there is no simple graph onn≥2vertices whose Laplacian spectrum is given by(0,1,…,n−1).” We provide a characterization of threshold graphs using the spectra of quotient matrices of its G-join graphs. As a consequence, we prove that given any n−1 positive integers λ2,…,λn such that λ2≤⋯≤λn, the n-tuple (0,λ2,…,λn) is realizable by the Laplacian matrix of a multidigraph. In particular, we show that (0,1,…,n−1) can be realizable by the Laplacian matrix of a multidigraph.

Laplacian spectra of cographs: A twin reduction perspective

A graph is a cograph if and only if it has no induced path on 4 vertices. The twin reduction graph of a graph G is denoted by RG. In this article, we describe the Laplacian eigenvalues and eigenvectors of a cograph G using its twin reduction graph RG, and characterize cographs with even and odd integer eigenvalues, respectively. We provide a construction for the Laplacian cospectral cographs. Further, we provide a complete description of the Laplacian spectrum of H-join of graphs when H is a cograph, and then obtain bounds for the algebraic connectivity of such graphs. We also provide some interesting observations on Hadamard diagonalizable cographs.

SYMMETRICAL MATRICES CHARACTERISTIC OF INTEGERS WITH INTEGER EIGEN VALUE

Estes (1992) stated that the set of eigen values from symmetrical matrices of Z is a set of totally real algebraic integers. Estes was not able to ensure that eigen values of a symmetrical matrices are integers. Mckee and Smyth (2007) observed more about the eigen value of symmetrical integer matrices. James and Chris proved that symmetrical integer matrices have eigen value with interval ranging in [-2,2]. Contrary to that, Martin and Wong (2009), stated that almost all integer matrices have no integer eigen value. Previous studies that could not show the characteristic of the eigen value made Cao and Koyunco studied and tried to determine the characteristic of symmetrical integer matrices for rank 2 and rank 3. The result shows that they have integer eigen value. In accordance to Cao and Kuyonco study, this article elaborates the characteristic of a symmetrical integer matrices for rank 4, and 5 to show the characteristic of a symmetrical integer matrices with integer eigen value for rank 1, 2, 3 and for rank 4 and 5.

Integral mixed circulant graphs

A mixed graph is said to be integral if all the eigenvalues of its Hermitian adjacency matrix are integers. The mixed circulant graphCirc(Zn,C) is a mixed graph on the vertex set Zn and edge set {(a,b):b−a∈C}, where 0∉C⊂Zn. If C is closed under inverse, then Circ(Zn,C) is called a circulant graph. We express the eigenvalues of Circ(Zn,C) in terms of primitive n-th roots of unity, and find a sufficient condition for integrality of the eigenvalues of Circ(Zn,C). For n≡0(mod4), we factorize the n-th degree cyclotomic polynomial into two irreducible factors over Q(i). Using this factorization, we characterize integral mixed circulant graphs in terms of their symbol set. We also express the integer eigenvalues of an integral oriented circulant graph in terms of a Ramanujan type sum, and discuss some of their properties.

Total Graphs Are Laplacian Integral

We prove that the Laplacian matrix of the total graph of a finite commutative ring with identity has integer eigenvalues and present a recursive formula for computing its eigenvalues and eigenvectors. We also prove that the total graph of a finite commutative local ring with identity is super integral and give an example showing that this is not true for arbitrary rings.

Dynamically Incoherent Surface Endomorphisms

We explicitly construct a dynamically incoherent partially hyperbolic endomorphisms of {{,mathrm{mathbb {T}},}}^2 in the homotopy class of any linear expanding map with integer eigenvalues. These examples exhibit branching of centre curves along countably many circles, and thus exhibit a form of coherence that has not been observed for invertible systems.

Minimal model for Hilbert space fragmentation with local constraints

Motivated by previous works on a Floquet version of the PXP model [Mukherjee {\it et al.} Phys. Rev. B 102, 075123 (2020), Mukherjee {\it et al.} Phys. Rev. B 101, 245107 (2020)], we study a one-dimensional spin-$1/2$ lattice model with three-spin interactions in the same constrained Hilbert space (where all configurations with two adjacent $S^z=\uparrow$ spins are excluded). We show that this model possesses an extensive fragmentation of the Hilbert space which leads to a breakdown of thermalization upon unitary evolution starting from a large class of simple initial states. Despite the non-integrable nature of the Hamiltonian, many of its high-energy eigenstates admit a quasiparticle description. A class of these, which we dub as "bubble eigenstates", have integer eigenvalues (including mid-spectrum zero modes) and strictly localized quasiparticles while another class contains mobile quasiparticles leading to a dispersion in momentum space. Other anomalous eigenstates that arise due to a {\it secondary} fragmentation mechanism, including those that lead to flat bands in momentum space due to destructive quantum interference, are also discussed. The consequences of adding a (non-commuting) staggered magnetic field and a PXP term respectively to this model, where the former preserves the Hilbert space fragmentation while the latter destroys it, are discussed. A Floquet version with time-dependent staggered field also evades thermalization with additional features like freezing of exponentially many states at special drive frequencies. Finally, we map the model to a $U(1)$ lattice gauge theory coupled to dynamical fermions and discuss the interpretation of some of these anomalous states in this language. A class of gauge-invariant states show reduced mobility of the elementary charged excitations with only certain charge-neutral objects being mobile suggesting a connection to fractons.

Virasoro algebras, kinematic space and the spectrum of modular Hamiltonians in CFT2

We construct an infinite class of eigenmodes with integer eigenvalues for the Vacuum Modular Hamiltonian of a single interval N in 2d CFT and study some of its interesting properties, which includes its action on OPE blocks as well as its bulk duals. Our analysis suggests that these eigenmodes, like the OPE blocks have a natural description on the so called kinematic space of CFT2 and in particular realize the Virasoro algebra of the theory on this kinematic space. Taken together, our results hints at the possibility of an effective description of the CFT2 in the kinematic space language.

Integral complete multipartite graphs

We give counterexamples to a result in F. Esser and F. Harary [2, Theorem 3] asserting that two nonisomorphic complete r-partite graphs with the same number of vertices have different spectral radii. We then derive some results on invariant factors and apply them to obtain relationship between the parameters of integral complete multipartite graphs and their integer eigenvalues. Necessary conditions for complete multipartite graphs to be integral are obtained.

Higher codimensional foliations with Kupka singularities

We consider holomorphic foliations of dimension [Formula: see text] and codimension [Formula: see text] in the projective space [Formula: see text], with a compact connected component of the Kupka set. We prove that if the transversal type is linear with positive integer eigenvalues, then the foliation consists of the fibers of a rational fibration [Formula: see text]. As a corollary, if [Formula: see text] and has a transversal type diagonal with different eigenvalues, then the Kupka component [Formula: see text] is a complete intersection and the leaves of the foliation define a rational fibration. The same conclusion holds if the Kupka set has a radial transversal type. Finally, as an application, we find a normal form for non-integrable codimension-one distributions on [Formula: see text].

The coprime quantum chain

In this paper we introduce and study the coprime quantum chain, i.e. a strongly correlated quantum system defined in terms of the integer eigenvalues ni of the occupation number operators at each site of a chain of length M. The ni’s take value in the interval [2,q] and may be regarded as Sz eigenvalues in the spin representation j = (q − 2)/2. The distinctive interaction of the model is based on the coprimality matrix : for the ferromagnetic case, this matrix assigns lower energy to configurations where occupation numbers ni and ni+1 of neighbouring sites share a common divisor, while for the anti-ferromagnetic case it assigns a lower energy to configurations where ni and ni+1 are coprime. The coprime chain, both in the ferro and anti-ferromagnetic cases, may present an exponential number of ground states whose values can be exactly computed by means of graph theoretical tools. In the ferromagnetic case there are generally also frustration phenomena. A fine tuning of local operators may lift the exponential ground state degeneracy and, according to which operators are switched on, the system may be driven into different classes of universality, among which the Ising or Potts universality class. The paper also contains an appendix by Don Zagier on the exact eigenvalues and eigenvectors of the coprimality matrix in the limit .

Symmetric Integer Matrices Having Integer Eigenvalues

We provide characterization of symmetric integer matrices for rank at most 2 that have integer spectrum and give some constructions for such matrices of rank 3. We also make some connection between Hanlon’s conjecture and integer eigenvalue problem.

Perfect state transfer in Laplacian quantum walk

For a graph $G$ and a related symmetric matrix $M$, the continuous-time quantum walk on $G$ relative to $M$ is defined as the unitary matrix $U(t) = \exp(-itM)$, where $t$ varies over the reals. Perfect state transfer occurs between vertices $u$ and $v$ at time $\tau$ if the $(u,v)$-entry of $U(\tau)$ has unit magnitude. This paper studies quantum walks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer: (1) If a $n$-vertex graph has perfect state transfer at time $\tau$ relative to the Laplacian, then so does its complement if $n\tau$ is an integer multiple of $2\pi$. As a corollary, the double cone over any $m$-vertex graph has perfect state transfer relative to the Laplacian if and only if $m \equiv 2 \pmod{4}$. This was previously known for a double cone over a clique (S. Bose, A. Casaccino, S. Mancini, S. Severini, Int. J. Quant. Inf., 7:11, 2009). (2) If a graph $G$ has perfect state transfer at time $\tau$ relative to the normalized Laplacian, then so does the weak product $G \times H$ if for any normalized Laplacian eigenvalues $\lambda$ of $G$ and $\mu$ of $H$, we have $\mu(\lambda-1)\tau$ is an integer multiple of $2\pi$. As a corollary, a weak product of $P_{3}$ with an even clique or an odd cube has perfect state transfer relative to the normalized Laplacian. It was known earlier that a weak product of a circulant with odd integer eigenvalues and an even cube or a Cartesian power of $P_{3}$ has perfect state transfer relative to the adjacency matrix. As for negative results, no path with four vertices or more has antipodal perfect state transfer relative to the normalized Laplacian. This almost matches the state of affairs under the adjacency matrix (C. Godsil, Discrete Math., 312:1, 2011).

The Laplace equation in 3D domains with cracks: dual singularities with log terms and extraction of corresponding edge flux intensity functions

The singular solution of the Laplace equation with a straight crack is represented by a series of eigenpairs, shadows, and their associated edge flux intensity functions (EFIFs). We address the computation of the EFIFs associated with the integer eigenvalues by the quasi‐dual function method (QDFM). The QDFM is based on the dual eigenpairs and shadows, and we exhibit the presence of logarithmic terms in the dual singularities associated with the integer eigenvalues. These are then used with the QDFM to extract EFIFs from p‐version finite element solutions. Numerical examples are provided. Copyright © 2015 John Wiley & Sons, Ltd.

THE INTEGRALITY OF AN ADAPTED PAIR

Let \( \mathfrak{a} \) be an algebraic Lie algebra. An adapted pair for \( \mathfrak{a} \) is a pair (h, η) consisting of an ad-semisimple element of h ∈ \( \mathfrak{a} \) and a regular element of η ∈ \( \mathfrak{a} \) ∗ satisfying (ad h)η = −η. In general such pairs are not easy to find and even more difficult to classify. A natural question is whether ad h has integer eigenvalues on \( \mathfrak{a} \), a property called the integrality of the adapted pair. In general this fails even for a Frobenius subalgebra of \( \mathfrak{s}\mathfrak{l} \)(3) and rather seriously in the sense that any rational number may serve as an eigenvalue. Nevertheless, integrality is shown to hold for any Frobenius Lie algebra which is a biparabolic subalgebra of a semisimple Lie algebra.Call a regular if there are no proper semi-invariant polynomial functions on \( \mathfrak{a}* \) and if the subalgebra of invariant functions is polynomial. In this case there are no known counter-examples to integrality. It is shown that if \( \mathfrak{a} \) is the canonical truncation of a biparabolic subalgebra of a simple Lie algebra \( \mathfrak{g} \) which is regular and admits an adapted pair (h, η), then the eigenvalues of ad h on \( \mathfrak{a} \) lie in (1/m)ℤ, where m is a coefficient of a simple root in the highest root of \( \mathfrak{g} \).Let \( \mathfrak{a} \) be a regular Lie algebra admitting an adapted pair (h, η). Let \( {\mathfrak{a}}_{\mathbf{\mathbb{Z}}} \) be the sub-algebra spanned by the eigensubspaces of ad h with integer eigenvalue. It is shown that the canonical truncation of \( {\mathfrak{a}}_{\mathbf{\mathbb{Z}}} \) is regular. Sufficient knowledge of the relation between the generators for the semi-invariant polynomial functions on \( \mathfrak{a}* \) and on \( {\mathfrak{a}}_{\mathbf{\mathbb{Z}}}^{*} \) can then lead to establishing that \( \mathfrak{a} \) = \( {\mathfrak{a}}_{\mathbf{\mathbb{Z}}} \). A particular interesting case is when a is the canonical truncation of a biparabolic subalgebra of a simple Lie algebra \( \mathfrak{g} \). If \( \mathfrak{g} \) is of type Α, then integrality already holds by the paragraph above. If \( \mathfrak{g} \) is of type C and \( \mathfrak{a} \) is a truncated parabolic subalgebra then a rather refined analysis shows that \( \mathfrak{a} \) = \( {\mathfrak{a}}_{\mathbf{\mathbb{Z}}} \). In principle this method can also be applied to biparabolic subalgebras in type C but there are some difficult combinatorial questions involving meanders to be resolved. Outside types A and C further technical complications arise out of an insufficient knowledge of the subalgebra of semi-invariant polynomial functions on the dual of a biparabolic.

Spanning trees and even integer eigenvalues of graphs

For a graph G, let L(G) and Q(G) be the Laplacian and signless Laplacian matrices of G, respectively, and τ(G) be the number of spanning trees of G. We prove that if G has an odd number of vertices and τ(G) is not divisible by 4, then (i) L(G) has no even integer eigenvalue, (ii) Q(G) has no integer eigenvalue λ≡2(mod4), and (iii) Q(G) has at most one eigenvalue λ≡0(mod4) and such an eigenvalue is simple. As a consequence, we extend previous results by Gutman and Sciriha and by Bapat on the nullity of adjacency matrices of the line graphs. We also show that if τ(G)=2ts with s odd, then the multiplicity of any even integer eigenvalue of Q(G) is at most t+1. Among other things, we prove that if L(G) or Q(G) has an even integer eigenvalue of multiplicity at least 2, then τ(G) is divisible by 4. As a very special case of this result, a conjecture by Zhou et al. (2013) on the nullity of adjacency matrices of the line graphs of unicyclic graphs follows.

Integral Cayley Graphs and Groups

We solve two open problems regarding the classification of certain classes of Cayley graphs with integer eigenvalues. We first classify all finite groups that have a nontrivial Cayley graph with integer eigenvalues, thus solving a problem proposed by Abdollahi and Jazaeri. The notion of Cayley integral groups was introduced by Klotz and Sander. These are groups for which every Cayley graph has only integer eigenvalues. In the second part of the paper, all Cayley integral groups are determined.

Spectrum of the [formula omitted]-Neumann Laplacian on the Fock space

The spectrum of the ∂¯-Neumann Laplacian on the Fock space L2(Cn,e−|z|2) is explicitly computed. It turns out that it consists of positive integer eigenvalues, each of which is of infinite multiplicity. Spectral analysis of the ∂¯-Neumann Laplacian on the Fock space is closely related to Schrödinger operators with magnetic fields and to the complex Witten Laplacian.