Infinite Class Research Articles

Revisiting and Improving Upper Bounds for Identifying Codes

An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. These codes have been widely studied for over two decades. We give an improvement over all the best known upper bounds, some of which have stood for over 20 years, for identifying codes in trees, proving the upper bound of $(n+\ell)/2$, where $n$ is the order and $\ell$ is the number of leaves (pendant vertices) of the graph. In addition to being an improvement in size, the new upper bound is also an improvement in generality, as it actually holds for bipartite graphs having no twins (pairs of vertices with the same closed or open neighbourhood) of degree 2 or greater. We also show that the bound is tight for an infinite class of graphs and that there are several structurally different families of trees attaining the bound. We then use our bound to derive a tight upper bound of $2n/3$ for twin-free bipartite graphs of order $n$, and characterize the extremal examples, as $2$-corona graphs of bipartite graphs. This is best possible, as there exist twin-free graphs, and trees with twins, that need $n-1$ vertices in any of their identifying codes. We also generalize the existing upper bound of $5n/7$ for graphs of order $n$ and girth at least 5 when there are no leaves, to the upper bound $\frac{5n+2\ell}{7}$ when leaves are allowed. This is tight for the $7$-cycle $C_7$ and for all stars.

On the Hamiltonian Property Hierarchy of 3-Connected Planar Graphs

The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph $K_2$. The graph $G$ is prism-hamiltonian if the prism over $G$ has a Hamilton cycle. A good even cactus is a connected graph in which every block is either an edge or an even cycle and every vertex is contained in at most two blocks. It is known that good even cacti are prism-hamiltonian. Indeed, showing the existence of a spanning good even cactus has become the most common technique in proving prism-hamiltonicity. Špacapan [S. Špacapan. A counterexample to prism-hamiltonicity of 3-connected planar graphs. J. Combin. Theory Ser. B, 146:364--371, 2021] asked whether having a spanning good even cactus is equivalent to having a hamiltonian prism for 3-connected planar graphs. In this article we answer his question in the negative, by showing that there are infinitely many 3-connected planar prism-hamiltonian graphs that have no spanning good even cactus. In addition, we prove the existence of an infinite class of 3-connected planar graphs that have a spanning good even cactus but no spanning good even cactus with maximum degree three.

Eigenvalue localization and Geršgorin disc-related problems on distance and distance-related matrices of graphs

The distance, distance signless Laplacian and distance Laplacian matrix of a simple connected graph [Formula: see text] are denoted by [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] is the diagonal matrix of vertex transmission. Ger[Formula: see text]gorin discs for any [Formula: see text] square matrix [Formula: see text] are the discs [Formula: see text], where [Formula: see text]. The famous Ger[Formula: see text]gorin disc theorem says that all the eigenvalues of a square matrix lie in the union of the Ger[Formula: see text]gorin discs of that matrix. In this paper, some classes of graphs are studied for which the smallest Ger[Formula: see text]gorin disc contains every distance and distance signless Laplacian eigenvalues except the spectral radius of the corresponding matrix. For all connected graphs, a lower bound and for trees, an upper bound of every distance signless Laplacian eigenvalues except the spectral radius is given in this paper. These bounds are comparatively better than the existing bounds. By applying these bounds, we find some infinite classes of graphs for which the smallest Ger[Formula: see text]gorin disc contains every distance signless Laplacian eigenvalues except the spectral radius of the distance signless Laplacian matrix. For the distance Laplacian eigenvalues, we have given an upper bound and then find a condition for which the smallest Ger[Formula: see text]gorin disc contains every distance Laplacian eigenvalue of the distance Laplacian matrix. These results give partial answers from some questions that are raised in [2].

Semiclassics of three-dimensional SCFTs from holography

We use holography to compute the large-N effective field theory along the moduli space of vacua of an infinite class of three-dimensional mathcal{N} = 2 SCFTs admitting a dual M-theory description. We focus in particular on toric models and show how the spectrum of large R-charge SCFT chiral scalar operators corresponds to a set of explicit semiclassical solutions of our effective field theory, which describe bound states of backreacting giant gravitons and baryonic-like M5-branes. Our semiclassical description allows for a direct computation of the scaling dimensions of these operators and provides a starting point for a semiclassical investigation of the SCFT data in the large R-charge sector. We consider the models corresponding to the Y12(ℙ2) and Q111 Sasaki-Einstein spaces as explicit examples.

Reconstructing homospectral inflationary potentials

Purely geometrical arguments show that there exist classes of homospectral inflationary cosmologies, i.e. different expansion histories producing the same spectrum of comoving curvature perturbations. We develop a general algorithm to reconstruct the potential of minimally-coupled single scalar fields from an arbitrary expansion history. We apply it to homospectral expansion histories to obtain the corresponding potentials, providing numerical and analytical examples. The infinite class of homospectral potentials depends on two free parameters, the initial energy scale and the initial value of the field, showing that in general it is impossible to reconstruct a unique potential from the curvature spectrum unless the initial energy scale and the field value are fixed, for instance through observation of primordial gravitational waves.

(S_2)-condition and Cohen–Macaulay binomial edge ideals

We describe the simplicial complex Delta such that the initial ideal of the binomial edge ideal J_textrm{G} of G is the Stanley-Reisner ideal of Delta . By using Delta we show that if J_textrm{G} is (S_2), then G is accessible. We also characterize all accessible blocks with whiskers of cycle rank 3 and we define a new infinite class of accessible blocks with whiskers for any cycle rank. Finally, by using a computational approach, we show that the graphs with at most 12 vertices whose binomial edge ideal is Cohen–Macaulay are all and only the accessible ones.

Propagating-wave approximation in two-dimensional potential scattering

We introduce a nonperturbative approximation scheme for performing scattering calculations in two dimensions that involves neglecting the contribution of the evanescent waves to the scattering amplitude. This corresponds to replacing the interaction potential $v$ with an associated energy-dependent nonlocal potential ${\mathscr{V}}_k$ that does not couple to the evanescent waves. The scattering solutions $\psi(\mathbf{r})$ of the Schr\"odinger equation, $(-\nabla^2+{\mathscr{V}}_k)\psi(\mathbf{r})=k^2\psi(\mathbf{r})$, has the remarkable property that their Fourier transform $\tilde\psi(\mathbf{p})$ vanishes unless $\mathbf{p}$ corresponds to the momentum of a classical particle whose magnitude equals $k$. We construct a transfer matrix for this class of nonlocal potentials and explore its representation in terms of the evolution operator for an effective non-unitary quantum system. We show that the above approximation reduces to the first Born approximation for weak potentials, and similarly to the semiclassical approximation, becomes valid at high energies. Furthermore, we identify an infinite class of complex potentials for which this approximation scheme is exact. We also discuss the appealing practical and mathematical aspects of this scheme.

An infinite class of Neumaier graphs and non-existence results

A Neumaier graph is a non-complete edge-regular graph containing a regular clique. A Neumaier graph that is not strongly regular is called a strictly Neumaier graph. In this work we present a new construction of strictly Neumaier graphs, and using Jacobi sums, we show that our construction produces infinitely many instances. Moreover, we prove some necessary conditions for the existence of (strictly) Neumaier graphs that allow us to show that several parameter sets are not admissible.

On a result of Koecher concerning Markov–Apéry-type formulas for the Riemann zeta function

Koecher in 1980 derived a method for obtaining identities for the Riemann zeta function at odd positive integers, including a classical result for [Formula: see text] due to Markov and rediscovered by Apéry. In this paper, we extend Koecher’s method to a very general setting and prove two more specific but still rather general results. As applications, we obtain infinite classes of identities for alternating Euler sums, further Markov–Apéry-type identities, and identities for even powers of [Formula: see text].

Yangian Symmetry in Holographic Correlators.

We point out that an infinite class of Witten diagrams is invariant under a Yangian symmetry. These diagrams are building blocks of holographic correlators and are related by a web of differential recursion relations. We show that Yangian invariance is equivalent to the consistency conditions of the recursion relations.

State polytopes related to two classes of combinatorial neural codes

Combinatorial neural codes are collections of 0/1 vectors that are used to model the co-firing patterns of a set of place cells in the brain. One wide-open problem in this area is to determine when a given code can be algorithmically drawn on the plane as a Venn diagram-like figure. A sufficient condition to do so is for the code to have a property called 2-inductively pierced. Gross, Obatake, and Youngs recently used toric algebra to show that a code on three neurons is 1-inductively pierced if and only if the toric ideal is trivial or generated by quadratics. No result is known for additional neurons in the same generality, part of the difficulty coming from the large number of codewords possible when additional neurons are used.In this article, we study two infinite classes of combinatorial neural codes in detail. For each code, we explicitly compute its universal Gröbner basis. This is done for the first class by recognizing that the codewords form a Lawrence-type matrix. With the second class, this is done by showing that the matrix is totally unimodular. These computations allow one to compute the state polytopes of the corresponding toric ideals, from which all distinct initial ideals may be computed efficiently. Moreover, we show that the state polytopes are combinatorially equivalent to well-known polytopes: the permutohedron and the stellohedron.

Construction of Dual-CISTs on an Infinite Class of Networks

The main method to achieve fault-tolerant network systems is by exploiting and effectively utilizing the edge-disjoint and/or inner-vertex-disjoint paths between pairs of source and destination vertices. Completely independent spanning trees (CISTs for short) are powerful tools for reliable broadcasting/unicasting and secure message distribution. Particularly, it has been shown that two CISTs have an application on configuring a protection routing in IP networks, such as mobile ad hoc networks and relatively large (static) network topologies with scalability in [IEEE/ACM Trans. Netw., 27 (2019) 1112-1123]. Many results focus on CISTs in specific networks in the literature, however, few results are given on an infinite class of networks having common properties. In this article, we prove the existence of dual-CISTs in an infinite number of networks satisfying some Hamilton sufficient conditions. A unique algorithm to construct a CIST-partition is proposed, which can be applied to not only many kinds of networks, but our algorithm can also be implemented very easily in parallel or distributed systems satisfying the conditions. In addition, we make a comparative analysis between the proposed conditions and several known results on an infinite number of networks, the advantage of our result is significant. In particular, the bound in our conditions is sharp. The results will provide a powerful framework for the design of fault-tolerant network topologies and routing protocols for future networks.

A Relation on Trees and the Topological Indices Based on Subgraph

A topological index reflects the physical, chemical and structural properties of a molecule, and its study has an important role in molecular topology, chemical graph theory and mathematical chemistry. It is a natural problem to characterize non-isomorphic graphs with the same topological index value. By introducing a relation on trees with respect to edge division vectors, denoted by $\langle\mathcal{T}_n, \preceq \rangle$, in this paper we give some results for the relation order in $\langle\mathcal{T}_n, \preceq \rangle$, it allows us to compare the size of the topological index value without relying on the specific forms of them, and naturally we can determine which trees have the same topological index value. Based on these results we characterize some classes of trees that are uniquely determined by their edge division vectors and construct infinite classes of non-isomorphic trees with the same topological index value, particularly such trees of order no more than $10$ are completely determined.

UV-complete gravity amplitudes and the triple product

We construct an infinite class of new ultraviolet-complete four-graviton scattering amplitudes that reduce to Einstein gravity at low energies, vanish at high energies, are meromorphic, and exhibit a triple-product structure ${\cal A}(s){\cal A}(t){\cal A}(u)$. The spectrum invariantly exhibits accumulation points in the form of infinite towers of states on each mass pole, whose residue can be positively expanded on tree-level exchanges of irreducible representations of the Lorentz group.

A New Class of Counting Distributions Embedded in the Lee–Carter Model for Mortality Projections: A Bayesian Approach

The Lee–Carter model, the dominant mortality projection modeling in the literature, was criticized for its homoscedastic error assumption. This was corrected in extensions to the model based on the assumption that the number of deaths follows Poisson or negative binomial distributions. We propose a new class of families of counting distributions, namely, the ABM class, which belongs to a wider class of natural exponential families. This class is characterized by its variance functions and contains the Poisson and the negative binomial distributions as special cases, offering an infinite class of additional counting distributions to be considered. We are guided by the principle that the choice of distribution should be made from a pool of distributions as large as possible. To this end, and following a data mining approach, a training set of historical mortality data of the population could be modeled using the ABM’s rich choice of distributions, and the chosen distribution should be the one that proved to offer superior projection results on a test set of mortality data. As an alternative to parameter estimation via the singular value decomposition used in the classical Lee–Carter model, we adopted Bayesian estimation, harnessing the Markov Chain Monte Carlo methodology. A numerical study demonstrates that when fitting mortality data using this new class of distributions, while traditional distributions may provide desirable projections for some populations, for others, alternative distributions within the ABM class can potentially produce superior results for the entire population or particular age groups, such as the oldest-old.

An update on the existence of Kirkman triple systems with steiner triple systems as subdesigns

AbstractA Kirkman triple system of order , KTS, is a resolvable Steiner triple system on elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS which contain as a subdesign a Steiner triple system of order , an STS. We present several different constructions for designs of this form. As a consequence, we completely settle the extremal case , for which a list of possible exceptions had remained for close to 30 years. Our new constructions also provide the first infinite classes for the more general problem. We reduce the other maximal case to (at present) three possible exceptions. In addition, we obtain results for other cases of the form and also near . Our primary method introduces a new type of Kirkman frame which contains group divisible design subsystems. These subsystems can occur with different configurations, and we use two different varieties in our constructions.

Poisson brackets and truncations in nonlinear reduced fluid models for plasmas

The Hamiltonian structure for an infinite class of nonlinear reduced fluid models, derived from a Hamiltonian drift-kinetic system, is explicitly provided in terms of the N+1 fluid moments evolving in each model of the class, with N an arbitrary positive integer. This improves previous results, in which the existence of the Hamiltonian structure was shown, but the complete explicit expression for the Poisson bracket of each model of the class was not provided. We also show that, whereas the Hamiltonian functional of the fluid models can be derived from that of the drift-kinetic system, by projecting the perturbation of the distribution function onto its truncated series in terms of Hermite polynomials, this is not the case for the Poisson bracket. Indeed, the antisymmetric bilinear form obtained by means of the aforementioned projection, although, interestingly, ”very similar” to the Poisson bracket of the fluid models, turns out to differ from it. The difference is found to reside in the coefficients W(N)lmn of the bilinear form, when the indices are such that l+m+n is even and l≥N+1,m≥N+1,n≥N+1. We show with a counterexample, related to the case N=2, that such bilinear form, in general, does not satisfy the Jacobi identity. We provide a physical interpretation of the set of variables G0,G1,…,GN, in terms of which the Poisson bracket of the fluid models exhibits a direct-sum structure, and point out an analogy between the present fluid reduction problem and the problem of the truncated quantum harmonic oscillator.

Simple Closed Quasigeodesics on Tetrahedra

Pogorelov proved in 1949 that every convex polyhedron has at least three simple closed quasigeodesics. Whereas a geodesic has exactly a π surface angle to either side at each point, a quasigeodesic has at most a π surface angle to either side at each point. Pogorelov’s existence proof did not suggest a way to identify the three quasigeodesics, and it is only recently that a finite algorithm has been proposed. Here we identify three simple closed quasigeodesics on any tetrahedron: at least one through one vertex, at least one through two vertices, and at least one through three vertices. The only exception is that isosceles tetrahedra have simple closed geodesics but do not have a 1-vertex quasigeodesic. We also identify an infinite class of tetrahedra that each have at least 34 simple closed quasigeodesics.

Using Model Theory to Find Decidable and Tractable Description Logics with Concrete Domains

Concrete domains have been introduced in the area of Description Logic to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. Unfortunately, in the presence of general concept inclusions (GCIs), which are supported by all modern DL systems, adding concrete domains may easily lead to undecidability. To regain decidability of the DL mathcal {ALC} in the presence of GCIs, quite strong restrictions, in sum called omega -admissibility, were imposed on the concrete domain. On the one hand, we generalize the notion of omega -admissibility from concrete domains with only binary predicates to concrete domains with predicates of arbitrary arity. On the other hand, we relate omega -admissibility to well-known notions from model theory. In particular, we show that finitely bounded homogeneous structures yield omega -admissible concrete domains. This allows us to show omega -admissibility of concrete domains using existing results from model theory. When integrating concrete domains into lightweight DLs of the mathcal {EL} family, achieving decidability is not enough. One wants reasoning in the resulting DL to be tractable. This can be achieved by using so-called p-admissible concrete domains and restricting the interaction between the DL and the concrete domain. We investigate p-admissibility from an algebraic point of view. Again, this yields strong algebraic tools for demonstrating p-admissibility. In particular, we obtain an expressive numerical p-admissible concrete domain based on the rational numbers. Although omega -admissibility and p-admissibility are orthogonal conditions that are almost exclusive, our algebraic characterizations of these two properties allow us to locate an infinite class of p-admissible concrete domains whose integration into mathcal {ALC} yields decidable DLs.

Mixed graphs with smallest eigenvalue greater than [formula omitted

The classical problem of characterizing the graphs whose eigenvalues lie in a given interval may date back to the work of J.H. Smith in 1970. Especially, the research on graphs with smallest eigenvalues not less than −2 has attracted widespread attention. Mixed graphs are natural generalizations of undirected graphs. In this paper, we completely characterize the mixed graphs with smallest Hermitian eigenvalue greater than −5+12. In fact, we found three infinite classes of mixed graphs and 30 scattered mixed graphs.