On automorphisms of a distance-regular graph with intersection array {39, 36, 1; 1, 2, 39}

Possible prime-order automorphisms and their fixed-point subgraphs are found for a hypothetical distance-regular graph with intersection array {39, 36, 1; 1, 2, 39}. It is shownthat graphs with intersection arrays {15, 12, 1; 1, 2, 15}, {35, 32, 1; 1, 2, 35}, and {39, 36, 1; 1, 2, 39} are not vertex-symmetric. © 2016, Pleiades Publishing, Ltd

On automorphisms of a distance-regular graph with intersection array {99, 84, 1; 1, 12, 99}

We find possible orders and fixed point subgraphs of a hypothetical distance-regular graph with intersection array {99, 84, 1; 1, 12, 99}. We show that, for a vertex-symmetric graph Γ with intersection array {99, 84, 1; 1, 12, 99}, its automorphism group is a {2, 3, 5}-group. © 2017, Pleiades Publishing, Ltd

Обратные задачи в классе Q-полиномиальных графов

In the class of distance-regular graphs Γ of diameter 3 with a pseudogeometric graph Γ3, feasible intersection arrays for the partial geometry were found for networks by Makhnev, Golubyatnikov, and Guo; for dual networks by Belousov and Makhnev; and for generalized quadrangles by Makhnev and Nirova. These authors obtained four infinite series of feasible intersection arrays of distance-regular graphs: {c2(u2 − m2) + 2c2m − c2 − 1, c2(u2 − m2), (c2 − 1)(u2 − m2) + 2c2m − c2; 1, c2, u2 − m2}, {mt, (t + 1)(m − 1), t + 1; 1, 1, (m − 1)t} for m ≤ t, {lt, (t − 1)(l − 1), t + 1; 1, t − 1, (l − 1)t}, and {a(p + 1), ap, a + 1; 1, a, ap}. We find all feasible intersection arrays of Q-polynomial graphs from these series. In particular, we show that, among these infinite families of feasible arrays, only two arrays ({7, 6, 5; 1, 2, 3} (folded 7-cube) and {191, 156, 153; 1, 4, 39}) correspond to Q-polynomial graphs. © 2020 Sverre Raffnsoe. All rights reserved.This work was supported by the Russian Foundation for Basic Research – the National Natural Science Foundation of China (project no. 20-51-53013_a)

Shilla Graphs with B=5 and B=6

A Q-polynomial Shilla graph with b=5 has intersection arrays {105t,4(21t+1),16(t+1);1,4(t+1),84t}, t∈{3,4,19}. The paper proves that distance-regular graphs with these intersection arrays do not exist. Moreover, feasible intersection arrays of Q-polynomial Shilla graphs with b=6 are found.This work was supported by RFBR and NSFC (project № 20-51-53013)

Открытые проблемы, сформулированные на XIII школе-конференции по теории групп, посвященной 85-летию В.А. Белоногова

A review of the main events of the 13th School–Conference on Group Theory, which was held online on August 3–7, 2020, is presented, and a list of open questions with comments is given. Open questions were formulated by the participants at the Open Problems Session held at the end of the school–conference. Among the posed problems there are a series of questions on the characterization of a finite group by its arithmetic invariants such as the spectrum, the Gruenberg–Kegel graph, the solvabile graph, and the degrees of irreducible complex characters (L. S. Kazarin, A. S. Kondrat’ev, and N. V. Maslova); the question of the conjugacy of the Sylow 2-subgroups in locally finite groups with additional conditions on these subgroups (V. D. Mazurov); a series of problems on the characterization of distance-regular graphs by their intersection arrays (A. A. Makhnev); the question of the nilpotent length of a finite solvable group whose Carter subgroup coincides with the Gaschütz subgroup (V. S. Monakhov); a series of problems about the structure of conjugately biprimitively finite groups or Shunkov groups (A. I. Sozutov); a question on the structure of some matrix groups over a residue ring Zn for a positive integer n (V. A. Roman’kov); a question on the characterization of Mazurov triples in finite groups (A. V. Timofeenko); and other open questions of the modern group theory and its applications. V. A. Belonogov’s brief biography and the list of his main publications are also presented. © Krasovskii Institute of Mathematics and Mechanics

Josephson array of mesoscopic objects. Modulation of system properties through the chemical potential

The phase diagram of a two-dimensional Josephson array of mesoscopic objects is examined. Quantum fluctuations in both the modulus and phase of the superconducting order parameter are taken into account within a lattice boson Hubbard model. Modulating the average occupation number $n_0$ of the sites in the system leads to changes in the state of the array, and the character of these changes depends significantly on the region of the phase diagram being examined. In the region where there are large quantum fluctuations in the phase of the superconducting order parameter, variation of the chemical potential causes oscillations with alternating superconducting (superfluid) and normal states of the array. On the other hand, in the region where the bosons interact weakly, the properties of the system depend monotonically on $n_0$. Lowering the temperature and increasing the particle interaction force lead to a reduction in the width of the region of variation in $n_0$ within which the system properties depend weakly on the average occupation number. The phase diagram of the array is obtained by mapping this quantum system onto a classical two-dimensional XY model with a renormalized Josephson coupling constant and is consistent with our quantum Path-Integral Monte Carlo calculations.Comment: 12 pages, 8 Postscript figure

Обратные задачи в теории графов: графы без треугольников

Graph r for a distance-regular graph r of diameter 3 can be strongly regular for i 2 or i = 3. Finding intersection array of graph r by t lio parameters of ri is an inverse problem. Earlier direct and inverse problems have been solved by A.A. Makhnev, M.S. Nirova for i = 3 and by A.A. Makhnev and D.V. Paduchikh for i = 2. In this work it is consider the case when graph r3 is strongly regular without triangles and v < 100000. © 2021 Махнев А.А., Белоусов И.Н., Падучих Д.В. All Rights Reserved.Работа выполнена при финансовой поддержке РФФИ и ГФЕН Китая в рамках научного проекта № 20-51-53013

Society and the state in the pandemic context

Based on the importance of an unprecedented new reality, the authors were tasked with studying and analyzing the problems of interaction between the State and society in the context of the pandemic and the introduction of state measures to prevent the spread of viral infection, which to a degree or another limit the rights and freedoms of a person and citize