Abstract
The main paradigm of smoothed analysis on graphs suggests that for any large graph $G$ in a certain class of graphs, perturbing slightly the edge set of $G$ at random (usually adding few random edges to $G$) typically results in a graph having much “nicer” properties. In this work, we study smoothed analysis on trees or, equivalently, on connected graphs. Given an $n$-vertex connected graph $G$, form a random supergraph $G^*$ of $G$ by turning every pair of vertices of $G$ into an edge with probability $\frac{\varepsilon}{n}$, where $\varepsilon$ is a small positive constant. This perturbation model has been studied previously in several contexts, including smoothed analysis, small world networks, and combinatorics. Connected graphs can be bad expanders, can have a very large diameter, and can possibly contain no long paths. In contrast, we show that if $G$ is an $n$-vertex connected graph, then typically $G^*$ has edge expansion $\Omega(\frac{1}{\log n})$, diameter $O(\log n)$, and vertex expansion $\Ome...
Highlights
In this paper, we consider the following model of randomly generated graphs
We begin by describing our results regarding the expansion properties of perturbed connected graphs
We prove an even stronger bound on the edge expansion of connected subsets of a perturbed connected graph
Summary
We consider the following model of randomly generated graphs. We are given a fixed undirected graph G = (V, E) on n vertices. Let R be the set of edges added and consider the random graph. Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany This model can be viewed as a generalization of the classical Erdős-Rényi random graph, where one starts from an empty graph and adds edges between all possible pairs of vertices independently with a given probability. The edge-isoperimetric number of G ( known as the Cheeger constant), denoted c(G), is defined by. The Cheeger constant has been studied extensively as it is related to a host of combinatorial properties of the underlying graph. There is a strong connection between the Cheeger constant of G and the mixing time of the lazy random walk on G
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