Computing complexity measures of degenerate graphs

We show that the VC-dimension of a graph can be computed in time $n^{\log d+1} d^{O(d)}$, where $d$ is the degeneracy of the input graph. The core idea of our algorithm is a data structure to efficiently query the number of vertices that see a specific subset of vertices inside of a (small) query set. The construction of this data structure takes time $O(d2^dn)$, afterwards queries can be computed efficiently using fast M\"obius inversion. This data structure turns out to be useful for a range of tasks, especially for finding bipartite patterns in degenerate graphs, and we outline an efficient algorithms for counting the number of times specific patterns occur in a graph. The largest factor in the running time of this algorithm is $O(n^c)$, where $c$ is a parameter of the pattern we call its left covering number. Concrete applications of this algorithm include counting the number of (non-induced) bicliques in linear time, the number of co-matchings in quadratic time, as well as a constant-factor approximation of the ladder index in linear time. Finally, we supplement our theoretical results with several implementations and run experiments on more than 200 real-world datasets -- the largest of which has 8 million edges -- where we obtain interesting insights into the VC-dimension of real-world networks.Comment: Accepted for publication in the 18th International Symposium on Parameterized and Exact Computation (IPEC 2023

Classification of Topological Isomers: Knots, Links, Rotaxanes, etc.

Topological isomers, i.e., knots, catenanes, rotaxanes, pseudoknots, hook-and-ladder, and Möbius molecules have so far been left out from the isomer classification schemes discussed. To expand the classification schemes to include the topological molecules such as knots and links, questions about the number of components and the number of crossings are incorporated into the scheme. In the case of rotaxanes and pseudoknots, which are topologically trivial, a procedure making them not trivial is described. For the hook-and-ladder as the well as Möbius type of isomers, a procedure is given that allows their classification. All the new procedures are included into the new classification scheme in such a way that all questions about topology precede the ordinary question tree. In that way, a molecule is first classified as a topological object, and then classical questions about its structure are asked

Distinguishing graphs by their left and right homomorphism profiles

We introduce a new property of graphs called ‘q-state Potts unique-ness’ and relate it to chromatic and Tutte uniqueness, and also to ‘chromatic–flow uniqueness’, recently studied by Duan, Wu and Yu. We establish for which edge-weighted graphs H homomor-phism functions from multigraphs G to H are specializations of the Tutte polynomial of G, in particular answering a question of Freed-man, Lovász and Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from multigraphs G to H are specializations of the ‘edge elimination polynomial’ of Averbouch, Godlin and Makowsky and the ‘induced subgraph poly-nomial’ of Tittmann, Averbouch and Makowsky. Unifying the study of these and related problems is the notion of the left and right homomorphism profiles of a graph.Ministerio de Educación y Ciencia MTM2008-05866-C03-01Junta de Andalucía FQM- 0164Junta de Andalucía P06-FQM-0164

Erlangen Program at Large-2.5: Induced Representations and Hypercomplex Numbers

In the search for hypercomplex analytic functions on the half-plane, we review the construction of induced representations of the group G=SL(2,R). Firstly we note that G-action on the homogeneous space G/H, where H is any one-dimensional subgroup of SL(2,R), is a linear-fractional transformation on hypercomplex numbers. Thus we investigate various hypercomplex characters of subgroups H. The correspondence between the structure of the group SL(2,R) and hypercomplex numbers can be illustrated in many other situations as well. We give examples of induced representations of SL(2,R) on spaces of hypercomplex valued functions, which are unitary in some sense. Raising/lowering operators for various subgroup prompt hypercomplex coefficients as well. The paper contains both English and Russian versions. Keywords: induced representation, unitary representations, SL(2,R), semisimple Lie group, complex numbers, dual numbers, double numbers, Moebius transformations, split-complex numbers, parabolic numbers, hyperbolic numbers, raising/lowering operators, creation/annihilation operatorsComment: LaTeX2e; 17 pp + 13 pp of a source code; 5 EPS pictures in two Figures; v2: minor improvements and corrections; v3: a section on raising/lowering operators is added; v4: typos are fixed; v5: Introduction is added, open problems are expanded.v6: Russian translation is added, references areupdated, NoWeb and C++ source codes are added as ancillary files. arXiv admin note: substantial text overlap with arXiv:0707.402

Transverse Signature-Type Change in Singular Semi-Riemannian Manifolds

In the early eighties, Hartle and Hawking put forth that signature-type change may be conceptually interesting, paving the way to the so-called no-boundary proposal for the initial conditions for the universe. Such singularity-free universes have no beginning, but they do have an origin of time. This leads to considerations of signature-type changing spacetimes, wherein the “initially” Riemannian manifold, characterized by a positive definite metric, undergoes a signature-type change, ultimately transitioning into a Lorentzian universe without boundaries or singularities. A metric with such a signature-type change is inherently degenerate or discontinuous at the locus of the signature change. We present a coherent framework for signature-type changing manifolds characterized by a degenerate yet smooth metric. We adapt well-established Lorentzian tools and results to the signature-type changing scenario. Subsequently, we explore global issues, specifically those related to the causal structure, in singular semi-Riemannian manifolds. We introduce new definitions that carry unforeseen causal implications. A noteworthy consequence is the presence of locally closed time-reversing loops through each point on the hypersurface. By imposing the constraint of global hyperbolicity on the Lorentzian region, we demonstrate that throughout every point in M, there always exists a pseudo-timelike loop. Or put another way, there always exists a closed pseudo-timelike path in M around which the direction of time reverses, and a consistent designation of future-directed and past-directed vectors cannot be defined. Moreover, we present a method for converting any arbitrary Lorentzian manifold (M,g) into a transverse type-changing semi-Riemannian manifold \ensuremath{(M,\tilde{g})}. Then we establish the Transformation Theorem, asserting that, conversely under certain conditions, such a metric (M,\tilde{g}) can be obtained from a Lorentz metric g through the aforementioned transformation procedure

Quantum Chemical Studies of Ring Currents of Aromatic Molecules

Aromaticity - the delocalization of electrons along a closed atomic circuit - has its manifestations in the energetic, structural, electronic, and spectroscopic properties of molecules and in how they react with each other. This phenomenon is central in chemistry, and the history of chemists using the concept as an “intuition pump” to understand and design new molecules goes back to the 1800s when Kekulé first time came up with the snake-eating-its-tail model of benzene. Those days predate the discovery of electron and quantum mechanics, and the concept has evolved since. While the physics of chemistry is understood, the utility of intuitive concepts still remains. Science as of today is still a human business, and to most of us chemists the fluctuations of fermionic field, or their computational representation as tensors, don’t give much food for thought. In this PhD thesis, I present my research in which quantum chemical methods were used to study different types of aromatic compounds. The focus is on assessing their aromaticity by probing the ring currents of molecules - the net flow of electrons around when it’s placed in a magnetic field. Calculation of this magnetically induced current density and the bond currents yield an accurate measure for electron delocalization. The studied systems present different types of aromaticities and aromatic molecules: through-bond aromaticity in the substituent ring of benzene derivatives, the intricacies of current pathways in naphtalene-fused porphyrinoids and in copper coordination complexes, and the magnetic-field orientation dependence of aromaticity in gaudiene, a spherical aromatic, but not spherically aromatic compound. The presented results disprove old conclusions for some compounds and enrich the understanding of others. In addition, the thesis gives a brief overview of computational quantum chemistry, and a slightly deeper one on aromaticity, presenting the ups and downs of different methods used to assess it computationally, and taking the reader on a tour to the zoo of different types of aromatic compounds.Kemiassa aromaattisuuden käsite ei tyypillisesti viittaa tuoksuihin, vaikka sen juuret ulottuvatkin näiden eriskummallisten molekyylien aromiin. Kemistit kutsuvat aromaattisuudeksi ilmiötä, jossa molekyylin elektronit eivät tyydy kohtaloonsa kahden atomiytimen välimaastossa, vaan leviävät yli atomikehikon muodostaen suljetun virtapiirin. Tuon syklisen delokalisaation myötä aromaattisilla molekyyleillä on erityisiä ominaisuuksia. Vastaavasti kuten elektroniikan virtapiirissä hieman vääränlainen resistori voi saada aikaan ennalta-arvaamattoman oikosulun, myös atomitasolla nämä näennäisen pienet muutokset voivat kytkeä aromaattisuuden pois päältä ja muuttaa molekyylin ominaisuuksia suuresti - kokonaisuus on enemmän kuin osiensa summa. Aromaattisia molekyylejä on kaikkialla, arkkityyppinä niistä on kuusikulmion muotoinen bentseeni. Evoluutio on valjastanut nämä aromaattiset molekyylit osaksi olevaisuuttamme: geeniperimämme on kirjoitettu DNA:n vakailla aromaattisilla emäspareilla. Solujemme toiminta pyörii aromaattisuutensa keinoin elektroneja välittävillä koentsyymeillä, ja aromaattisuudella on mitä keskeisin rooli porfyriinimolekyylien kyvyssä niin sitoa happea kuljettavat rauta-atomit verisoluissamme kuin vastaanottaa auringon säteilemä energia kasvien viherhiukkasissa. Luonto on löytänyt aromaattisille molekyyleille paljon käyttöä, ja näistä prosesseista kummunneet kemistit jatkavat sen työtä. Aromaattisuuden käsite on tärkeä väline tässä työkalupakissa. Toinen tärkeä työkalu on kvanttifysiikka - sen avulla olemme noin sadan vuoden ajan kyenneet kunnolla ymmärtämään molekyylejä, sekä sittemmin tietokoneiden kehittymisen myötä onnistuneet laskemaan niiden ominaisuuksia tarkasti. Väitöstutkimuksessani sovelsin laskennallisen kvanttikemian menetelmiä aromaattisuuden määrittämiseksi. Väitöskirja rakentuu neljän tieteellisen julkaisun ympärille, joissa tarkastelemme erilaisten molekyylien aromaattisuutta laskemalla rengasvirtoja - ulkoisen magneettikentän aiheuttamaa elektronien virtausta atomiydinten muodostaman virtapiirin ympäri. Tulosten avulla kykenimme ymmärtämään tiettyjen molekyylien aromaattisuutta paremmin sekä kumoamaan joitakin väärinkäsityksiä. Tehdyn perustutkimuksen löydökset todentavat rengasvirtojen antavan tarkan ja fysikaalisesti perustellun kuvan aromaattisuudesta - erityisesti jos sitä vertaa 1800-luvun hajunvaraiseen toimintaan - ja rengasvirtojen olevan tärkeä menetelmä aromaattisuuden ymmärtämisessä