On the probability of connectedness

نویسنده

  • Peter J. Cameron
چکیده

Given a class of structures with a notion of connectedness (satisfying some reasonable assumptions), we consider the limit (as n -+ oc) of the probability that a random (labelled or unlabelled) n-element structure in the class is connected. The paper consists of three parts: two specific examples, N-free graphs and posets, where the limiting probability of connectedness is one-half and the golden ratio respectively; an investigation of the relation between this question and the growth rate of the number of structures in the class; and a generalisation of the problem to other combinatorial constructions motivated in part by the group-theoretic constructions of direct and wreath product. 1. An example: N-free posets In this section, certain graphs and posets will be considered. The notion of connectedness of a graph is clear, while a poset is connected if its comparability graph is connected. Given a class ~ of graphs or posets, we select at random an isomorphism class of n-element structures in g (the unlabelled case) or a structure on {1,2 . . . . . n} isomorphic to a structure in cg (the labelled case); let p,, be the probability that the chosen structure is connected. Our general question is: Does lim . . . . (Pn) exist, and i f so what is" it? The symbol N will denote the graph or the poset which is shown in Fig. 1. A graph or poser is called N-free if it does not contain N as an induced substructure. The class of N-free graphs has been studied in many contexts, under many different names. I summarise the main facts. The complement of an N-free graph is N-free. An N-free graph with more than one vertex is connected if and only if its complement is disconnected. * E-mail: [email protected]. 0012-365X/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved PH SOOI2-365X(96)O0226-9 176 P.J. Cameron / Discrete Mathematics 167/168 (1997) 175 187

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Relationship of Hope and Life Satisfaction with Students Bullying and Bullying Victimization with the Mediating Role of School Connectedness: A Path Analysis Model

Background and Aim: The aim of the present study was to investigate the relationship of hope and life satisfaction with students bullying and bullying victimization with the mediating role of school connectedness.                                     Materials and Methods: This was a cross-sectional study, the statistical population being high school students of Tehran city in the school year 2...

متن کامل

Investigation of a Preliminary Mixed Method of Phubbing and Social Connectedness in Adolescents

Background: The exponential growth of smartphones has afforded many users with ubiquitous access to socialization as seen in the various mobile apps used to communicate and connect with others. The present study employed mixed-method approaches to analyse the impact of phubbing on social connectedness among adolescents in Malaysia. Methods:</stron...

متن کامل

Uniform connectedness and uniform local connectedness for lattice-valued uniform convergence spaces

We apply Preuss' concept of $mbbe$-connectedness to the categories of lattice-valued uniform convergence spaces and of lattice-valued uniform spaces. A space is uniformly $mbbe$-connected if the only uniformly continuous mappings from the space to a space in the class $mbbe$ are the constant mappings. We develop the basic theory for $mbbe$-connected sets, including the product theorem. Furtherm...

متن کامل

Somewhat-connectedness and somewhat-continuity in the product space

In this paper, the concept of somewhat-connected space will be introduced and characterized. Its connection with the other well-known concepts such as the classical connectedness, the $omega_theta$-connectedness, and the $omega$-connectedness will be determined. Moreover, the concept of textit{somewhat}-continuous function from an arbitrary topological space into the product space will be chara...

متن کامل

A Model of Architectural Factors Influence on Students' Perception of School Climate for Providing School Connectedness

Background and Objectives: The aim of this research was to introduce architectural factors, influencing students' perception of school climate, having a great effect on students' achievement academically, mentally and physically, one of the problems of high school students is their unwillingness for learning and passes their times at educational setting. School connectedness has an effect on st...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • Discrete Mathematics

دوره 167-168  شماره 

صفحات  -

تاریخ انتشار 1997