## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, August 26, 2014, 12:15 pm

Duration: 30 minutes

Location: LFW C4

Speaker: Konstantinos Panagiotou (Mathematisches Institut der Universität München)

## Scaling Limits for Random Graphs

Given a connected graph G with vertex set V we can associate naturally to it a metric space (V, d), where d(u,v) denotes the shortest path distance of u and v in G. If G is a random graph, then this metric space is itself a random variable, and the aim of this talk is to study its asymptotic properties when the size of G becomes large.

We consider so-called block-stable classes that can be defined as follows. Suppose that we are given a class B of 2-connected graphs, which may also include the graph consisting of a single edge. Then we let C = C(B) be the class of all connected graphs whose blocks, i.e., maximal subgraphs that contain no cut-vertex, are in B. For example, if B is the class of all 2-connected planar graphs (and the single edge), then C is the class of all connected planar graphs; if B is the class that contains only the graph that consists of a single edge, then we recover the class of trees.

For a random graph from a block-stable class with n vertices we study the associated random metric space. For a general class of choices for B we show that it converges (in a well-defined sense) to an object called the continuum random tree, and obtain as a corollary for example the distribution of the diameter. In this talk I will give an informal picture of this object and describe the intuition behind its seminal construction by Aldous.

Joint work with B. Stufler and K. Weller.

Information for students and suggested topics for student talks

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