Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar Talk Information |

**Date and Time**: Tuesday, November 28, 2006, 12:15 pm

**Duration**: This information is not available in the database

**Location**: OAT S15/S16/S17

**Speaker**: Stefanie Gerke

We call a non-empty class A of labelled graphs addable if for each graph G in A and any two vertices u and v in distinct components of G, the graph obtained by adding the edge {u,v} to G is in A. Examples of addable graph classes include forests, planar graphs, and triangle-free graphs. The set of graphs in A with vertex set {1,..,n} is denoted by A_n. McDiarmid, Steger, and Welsh conjecture that for an addable class of graphs with the property that A_n for all sufficiently large n, an element R_n drawn uniformly at random from A_n satisfies \liminf_{n\rightarrow \infty} P[R_n is connected] >= e^{-1/2}. Let us remark that one cannot increase e^{-1/2} as the class of forests shows.

McDiarmid, Steger, and Welsh proved the conjecture when e^{-1/2} is replaced by e^{-1}. We improve this constant to e^{-0.7983}. To prove this result we find for a>0, an upper bound on the generalized Randic index R_-a(T) of a tree T, that is, the sum over all edges {u,v} in E(T) of (d(u)d(v))^{-a}, where d(u) is the degree of u in T. Randic introduced this measure to give a theoretical characterization of molecular branching. For every a<0, we find an effectively computable constant b=b(a) such that for all trees T on n>2 vertices, R_{-a}(T)<= b(n+1). We also construct infinitely many trees such that R_{-a}(T)>= b(n-1).

Joint work with Paul Balister and Bela Bollobas.

Upcoming talks | All previous talks | Talks by speaker | Upcoming talks in iCal format (beta version!)

Previous talks by year: 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996

Information for students and suggested topics for student talks

Automatic MiSe System Software Version 1.4803M | admin login