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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2015, Volume 25, Issue 1, Pages 3–11 (Mi vuu459) MATHEMATICS

The graph of reflexive-transitive relations and the graph of finite topologies

Kh. Sh. Al' Dzhabriab

a Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia
b University of Al-Qadisiyah, ul. Babilon, 29, Al Diwaniyah, Iraq

Abstract: Any binary relation $\sigma\subseteq X^2$ (where $X$ is an arbitrary set) generates on the set $X^2$ a characteristic function: if $(x,y)\in\sigma$, then $\sigma(x,y)=1$, otherwise $\sigma(x,y)=0$. In terms of characteristic functions we introduce on the set of all binary relations of the set $X$ the concept of a binary reflexive relation of adjacency and determine an algebraic system consisting of all binary relations of the set and of all unordered pairs of various adjacent binary relations. If $X$ is a finite set then this algebraic system is a graph (“the graph of graphs”).
It is shown that if $\sigma$ and $\tau$ are adjacent relations then $\sigma$ is a reflexive-transitive relation if and only if $\tau$ is a reflexive-transitive relation. Several structure features of the graph $G(X)$ of reflexive-transitive relations are investigated. In particular, if $X$ consists of $n$ elements, and $T_0(n)$ is the number of labeled $T_0$-topologies defined on the set $X$, then the number of connected components is equal to $\sum_{m=1}^nS(n,m)T_0(m-1)$, where $S(n,m)$ are Stirling numbers of second kind. (It is well known that the number of vertices in a graph $G(X)$ is equal to $\sum_{m=1}^nS(n,m)T_0(m)$.)

Keywords: graph, reflexive-transitive relation, finite topology. Full text: PDF file (237 kB) References: PDF file   HTML file
UDC: 519.175+519.115.5
MSC: 05C30

Citation: Kh. Sh. Al' Dzhabri, “The graph of reflexive-transitive relations and the graph of finite topologies”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 25:1 (2015), 3–11 Citation in format AMSBIB
\Bibitem{Al 15} \by Kh.~Sh.~Al' Dzhabri \paper The graph of reflexive-transitive relations and the graph of finite topologies \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2015 \vol 25 \issue 1 \pages 3--11 \mathnet{http://mi.mathnet.ru/vuu459} \elib{http://elibrary.ru/item.asp?id=23142044} 

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• http://mi.mathnet.ru/eng/vuu/v25/i1/p3

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This publication is cited in the following articles:
1. Kh. Sh. Al Dzhabri, V. I. Rodionov, “Graf atsiklicheskikh orgrafov”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 25:4 (2015), 441–452  2. V. I. Rodionov, “O perechislenii chastichnykh poryadkov, opredelennykh na konechnom mnozhestve”, Sib. elektron. matem. izv., 13 (2016), 318–330  3. Kh. Sh. Al Dzhabri, V. I. Rodionov, “Ob opornykh mnozhestvakh atsiklicheskikh i tranzitivnykh orientirovannykh grafov”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:2 (2017), 153–161   •  Contact us: math-net2020_04 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2020