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Iraq; Russia Al Diwaniyah; Izhevsk
Section  Mathematics 
Title  The graph of reflexivetransitive relations and the graph of finite topologies 
Author(s)  Al' Dzhabri Kh.Sh.^{ab} 
Affiliations  Udmurt State University^{a}, University of AlQadisiyah^{b} 
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 reflexivetransitive relation if and only if $\tau$ is a reflexivetransitive relation. Several structure features of the graph $G(X)$ of reflexivetransitive 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 $\sum_{m=1}^nS(n,m) T_0(m1),$ 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, reflexivetransitive relation, finite topology 
UDC  519.175, 519.115.5 
MSC  05C30 
DOI  10.20537/vm150101 
Received  12 November 2014 
Language  Russian 
Citation  Al' Dzhabri Kh.Sh. The graph of reflexivetransitive relations and the graph of finite topologies, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 1, pp. 311. 
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