Dr Mikhail Isaev

Research Overview

Many enumeration problems in combinatorics, including such fundamental questions as the number of regular graphs, can be expressed as multidimensional complex integrals. In fact, the asymptotic behaviour can be found by concentrating the integral in a small region and then approximating the integrand inside that region. This idea forms the core of the complex-analytic approach pioneered by Prof. McKay and Prof. Wormald in 1990 which was subsequently used for enumeration of such important objects as graphs, hypergraphs, digraphs, tournaments, integer matrices, eulerian circuits, eulerian orientations, correlation-immune functions, design matrices. My main research area is devoted to developing a general theory for estimating such integrals that unifies and extends previously known ad-hoc methods. I also interested in properties of random graphs and hypergraphs with given degrees.

Selected Publications

[1] Isaev, Mikhail ; Iyer, Tejas ; McKay, Brendan D. "Asymptotic enumeration of orientations of a graph as a function of the out-degree sequence". In: Electronic Journal of Combinatorics. 2020 ; Vol. 27, No. 1. https://doi.org/10.37236/8929

[2] Altman, Daniel ; Greenhill, Catherine ; Isaev, Mikhail ; Ramadurai, Reshma. "A threshold result for loose Hamiltonicity in random regular uniform hypergraphs". In: Journal of Combinatorial Theory, Series B. 2020 ; Vol. 142. pp. 307-373. https://doi.org/10.1016/j.jctb.2019.11.001

[3] Isaev, Mikhail ; McKay, Brendan D. "Complex martingales and asymptotic enumeration". In: Random Structures & Algorithms. 2018 ; Vol. 52, No. 4. pp. 617-661. https://doi.org/10.1002/rsa.20754

[4] Greenhill, Catherine ; Isaev, Mikhail ; Kwan, Matthew ; McKay, Brendan Damien. "The average number of spanning trees in sparse graphs with given degrees". In: European Journal of Combinatorics. 2017 ; Vol. 63. pp. 6-25. https://doi.org/10.1016/j.ejc.2017.02.003

[5] Isaev, M. I. "Asymptotic enumeration of Eulerian circuits in graphs with strong mixing properties". In: Izvestiya: Mathematics. 2013 ; Vol. 77, No. 6. pp. 1105-1129. https://doi.org/10.1070/IM2013v077n06ABEH002671