Asymptotic Delsarte cliques in distance-regular graphs

We give a new bound on the parameter $\lambda$ (number of common neighbors of a pair of adjacent vertices) in a distance-regular graph $G$, improving and generalizing bounds for strongly regular graphs by Spielman (1996) and Pyber (2014). The new bound is one of the ingredients of recent progress on the complexity of testing isomorphism of strongly regular graphs (Babai, Chen, Sun, Teng, Wilmes 2013). The proof is based on a clique geometry found by Metsch (1991) under certain constraints on the parameters. We also give a simplified proof of the following asymptotic consequence of Metsch's result: if $k\mu = o(\lambda^2)$ then each edge of $G$ belongs to a unique maximal clique of size asymptotically equal to $\lambda$, and all other cliques have size $o(\lambda)$. Here $k$ denotes the degree and $\mu$ the number of common neighbors of a pair of vertices at distance 2. We point out that Metsch's cliques are "asymptotically Delsarte" when $k\mu = o(\lambda^2)$, so families of distance-regular graphs with parameters satisfying $k\mu = o(\lambda^2)$ are "asymptotically Delsarte-geometric."Comment: 10 page

On the automorphism groups of strongly regular graphs II

We derive strong constraints on the automorphism groups of strongly regular (SR) graphs, resolving old problems motivated by Peter Cameron's 1981 description of large primitive groups.Trivial SR graphs are the disjoint unions of cliques of equal size and their complements. Graphic SR graphs are the line-graphs of cliques and of regular bipartite cliques (complete bipartite graphs with equal parts) and their complements.We conjecture that the order of the automorphism group of a non-trivial, non-graphic SR graph is quasi-polynomially bounded, i.e., it is at most exp((logn)C) for some constant C, where n is the number of vertices.While the conjecture remains open, we find surprisingly strong bounds on important parameters of the automorphism group. In particular, we show that the order of every automorphism is O(n8), and in fact O(n) if we exclude the line-graphs of certain geometries. We prove the conjecture for the case when the automorphism group is primitive; in this case we obtain a nearly tight n1+log2n bound.We obtain these bounds by bounding the fixicity of the automorphism group, i.e., the maximum number of fixed points of non-identity automorphisms, in terms of the second largest (in magnitude) eigenvalue and the maximum number of pairwise common neighbors of a regular graph. We connect the order of the automorphisms to the fixicity through an old lemma by Ákos Seress and the author.We propose to extend these investigations to primitive coherent configurations and offer problems and conjectures in this direction. Part of the motivation comes from the complexity of the Graph Isomorphism problem

On the Nonuniform Fisher Inequality

AbstractLet F be a family of m subsets (lines) of a set of n elements (points). Suppose that each pair of lines has λ points in common for some positive λ. The Nonuniform Fisher Inequality asserts that under these circumstances m ⩽ n. We examine the case when m = n. We give a short proof of the fact that (with the exception of a trivial case) such an F must behave like a geometry in the following sense: a line must pass through each pair of points. This generalizes a result of de Bruijn and Erdös

Automorphism groups of graphs and edge-contraction

AbstractIf a class C of finite graphs is closed under contraction and forming subgraphs, and if every finite abstract group occurs as the automorphism group of some graph in C, then C contains all finite graphs (up to isomorphism). Also related results concerning automorphism groups of graphs on given surfaces are mentioned

Asymmetric coloring of locally finite graphs and profinite permutation groups: Tucker's Conjecture confirmed

An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph is the minimal degree of its automorphism group, i.e., the minimum number of elements that are moved (not fixed) by any non-identity automorphism. We confirm Tom Tucker's “Infinite Motion Conjecture” that connected locally finite graphs with infinite motion admit an asymmetric 2-coloring. We infer this from the more general result that the inverse limit of an infinite sequence of finite permutation groups with disjoint domains, viewed as a permutation group on the union of those domains, admits an asymmetric 2-coloring. The proof is based on the study of the interaction between epimorphisms of finite permutation groups and the structure of the setwise stabilizers of subsets of their domains. We note connections of the subject to computational group theory, asymptotic group theory, highly regular structures, and the Graph Isomorphism problem, and list a number of open problems. © 2021 Elsevier Inc

An exponential lower bound for Individualization-Refinement algorithms for Graph Isomorphism

The individualization-refinement paradigm provides a strong toolbox for testing isomorphism of two graphs and indeed, the currently fastest implementations of isomorphism solvers all follow this approach. While these solvers are fast in practice, from a theoretical point of view, no general lower bounds concerning the worst case complexity of these tools are known. In fact, it is an open question whether individualization-refinement algorithms can achieve upper bounds on the running time similar to the more theoretical techniques based on a group theoretic approach. In this work we give a negative answer to this question and construct a family of graphs on which algorithms based on the individualization-refinement paradigm require exponential time. Contrary to a previous construction of Miyazaki, that only applies to a specific implementation within the individualization-refinement framework, our construction is immune to changing the cell selector, or adding various heuristic invariants to the algorithm. Furthermore, our graphs also provide exponential lower bounds in the case when the $k$-dimensional Weisfeiler-Leman algorithm is used to replace the standard color refinement operator and the arguments even work when the entire automorphism group of the inputs is initially provided to the algorithm.Comment: 21 page

Sidon Sets in Groups and Induced Subgraphs of Cayley Graphs

Let S be a subset of a group G. We call S a Sidon subset of the first (second) kind, if for any x, y, z, w ∈ S of which at least 3 are different, xy ≠ zw (xy-1 ≠ zw-1, resp.). (For abelian groups, the two notions coincide.) If G has a Sidon subset of the second kind with n elements then every n-vertex graph is an induced subgraph of some Cayley graph of G. We prove that a sufficient condition for G to have a Sidon subset of order n (of either kind) is that (❘G❘ ⩾ cn3. For elementary Abelian groups of square order, ❘G❘ ⩾ n2 is sufficient. We prove that most graphs on n vertices are not induced subgraphs of any vertex transitive graph with <cn2/log2n vertices. We comment on embedding trees and, in particular, stars, as induced subgraphs of Cayley graphs, and on the related problem of product-free (sum-free) sets in groups. We summarize the known results on the cardinality of Sidon sets of infinite groups, and formulate a number of open problems.We warn the reader that the sets considered in this paper are different from the Sidon sets Fourier analysts investigate

Antiaritmiás és proaritmiás mechanizmusok elemzése = Analysis of antiarrhythmic and proarrhythmic mechanism

Szívizomsejtben a K+-áramok antiaritmiás ''repolarizációs tartalékot'' tartanak fenn: Az IKr, IKs és IK1 akár egyikének gátlása súlyos proaritmiás repolarizáció megnyúlást okoz és letális kamrafibrillációhoz vezethet. - Alloxán-diabeteses kutyán az Ito és IKs csökkenése ugyanilyen hatású, amit a Kv4.3 és Mink csatorna fehérje expresszió egyidejű gátoltsága kisér. - Szívbillyentyű átültetés céljából használt, egészséges donor szivekből izolált humán kamrai miocitákon az IKs csökkenése szintén növeli a proaritmia és a hirtelen szívhalál rizikóját. - A K+-csatorna alegység expresszió mértékének sorrendje: KvLQT1 és ERG1-nél: ember>nyúl>tengerimalad, Mink-nél: tengerimalac>ember>nyúl. - Az INa+/Ca2+ csereáram gátlása kutyán antiaritmiás hatású, csökkenti az utódepolarizációt és a triggerelt aktivitást. - A leghatékonyabb új antiaritmikumok ''többszörös (tedisamil) ill. ''hibrid'' (azimilid, terikalant) ioncsatorna-gátlók. A pitvarszelektiv antiaritmiás szerek hatásmódja: az IKAch, IKur és IK1 izolált blokkolása. - A prekondicionálás antiaritmiás hatásában kutyán a NO trigger és mediátor, prosztaciklin- és szabadgyök-képződés viszont nem vesz részt ebben. | In the myocardial cell, there is a 'repolarization reserve' maintained by K+ currents: inhibition any of the IKr, IKs and IK1 results in severe proarrhythmic prolongation of repolarization and may cause lethal ventricular fibrillation. - The decrease of IKs and also Ito exhibits the same effect in alloxan-diabetic dogs which is accompanied by concomitant retardation of the expression of the K+ channel proteins Kv4.3 and Mink. - In undiseased human ventricular myocites obtained from donors for valve transplant surgery diminution of IKs also increases the risk of proarrhythmia and suddenc cardiac death. The extent of the order of expression of K+ channel subunits: human>rabbit>guinea pig with KvLQT1 and ERG1, and guinea pig>human>rabbit with Mink. - Inhibition of the INa+/Ca2+ exchange current is antiarrhythmic in dogs; it decreases the afterdepolarization and triggered activity. - The most efficacious new antiarrhythmics are ''multiple'' (tedisamil) or 'hybrid' (azimilid, terikalant) ion-channel blockers. - The mode of action of atrial-selective antiarrhythmics: isolated block of IKAch or IKur or IKs. - In the antiarrhythmic action of preconditioning in dogs nitrogen oxide acts as trigger and mediator, whereas formation of prostacyclin and free radicals are not involved in this effect