We relate two important notions in graph theory: expanders which are highly connected graphs, and modularity a parameter of a graph that is primarily used in community detection. More precisely, we show that a graph having modularity bounded below 1 is equivalent to it having a large subgraph which is an expander. We further show that a connected component H will be split in an optimal partition of the host graph G if and only if the relative size of H in G is greater than an expansion constant of H. This is a further exploration of the resolution limit known for modularity, and indeed recovers the bound that a connected component H in the host graph G will not be split if e(H) < √{2e(G)}.

For a given graph G, modularity gives a score to each vertex partition, with higher values taken to indicate that the partition better captures community structure in G. The modularity q^*(G) (where 0 <= q^*(G)<= 1) of the graph G is defined to be the maximum over all vertex partitions of the modularity value. Given the prominence of modularity in community detection, it is an important graph parameter to understand mathematically. For the Erdös-Rényi random graph G_{n,p} with n vertices and edge-probability p, the likely modularity has three distinct phases. For np <= 1+o(1) the modularity is 1+o(1) with high probability (whp), and for np --> infty the modularity is o(1) whp. Between these regions the modularity is non-trivial: for constants 1 < c_0 <= c_1 there exists delta>0 such that when c_0 <= np <= c_1 we have delta<q^*(G)<1-delta whp. For this critical region, we show that whp q^*(G_{n,p}) has order (np)^{-1/2}, in accord with a conjecture by Reichardt and Bornholdt in 2006 (and disproving another conjecture from the physics literature).

A hyperplane search tree is a binary tree used to store a set S of n d-dimensional data points. In a random hyperplane search tree for S, the root represents a hyperplane defined by d data points drawn uniformly at random from S. The remaining data points are split by the hyperplane, and the definition is used recursively on each subset. We assume that the data are points in general position in $\mathbb{R}^d$. We show that, uniformly over all such data sets S, the expected height of the hyperplane tree is not worse than that of the k-d tree or the ordinary one-dimensional random binary search tree, and that, for any fixed $d\ge3$, the expected height improves over that of the standard random binary search tree by an asymptotic factor strictly greater than one.

We are interested in a version of graph coloring where there is a "co-site" constraint value k. Given a graph G with a nonnegative integral demand x v at each node v, we must assign x v positive integers (colors) to each node v such that the same integer is never assigned to adjacent nodes, and two distinct integers assigned to a single node differ by at least k. The aim is to minimize the span, that is, the largest integer assigned to a node. This problem is motivated by radio channel assignment where one has to assign frequencies to transmitters so as to avoid interference. We compare the span with a clique-based lower bound when some of the demands are large. We introduce the relevant graph invariant, the k-imperfection ratio, give equivalent definitions, and investigate some of its properties. The k-imperfection ratio is always at least 1: we call a graph k-perfect when it equals 1. Then 1-perfect is the same as perfect, and we see that for many classes of perfect graphs, each graph in the class is k-perfect for all k. These classes include bipartite graphs and more generally comparability graphs, co-comparability graphs, and line-graphs of bipartite graphs.

In cellular telephone networks, sets of radio channels (colors) must be assigned to transmitters (vertices) while avoiding interference. Often, the transmitters are laid out like vertices of a triangular lattice in the plane. We investigated the corresponding weighted coloring problem of assigning sets of colors to vertices of the triangular lattice so that the sets of colors assigned to adjacent vertices are disjoint. We present a hardness result and an efficient algorithm yielding an approximate solution. © 2000 John Wiley & Sons, Inc.

The problem of finding a perfect matching of small total length in a complete graph whose vertices are points in the interval [0,1] is considered. The greedy heuristic for this problem repeatedly picks the two closest unmatched points x and y, and adds the edge $xy$ to the matching. It is shown that if $2n$ points are randomly chosen uniformly in $[0,1]$, then the expected length of the matching given by the greedy algorithm is $\theta (\log n)$. This compares unfavourably with the length of the shortest perfect matching, which is always less than 1.

Lower bounds are given for certain average search times in a set with a random linear order about which there is partial information. These bounds extend various recent worst-case and average-case results, in particular those of Alt and Mehlhorn, Borodin et al., and Mairson concerning searching “semi-sorted” tables and trade-offs between presorting time and search time. We make fuller use of the framework of information theory than have previous investigations.

Certain branch-and-bound algorithms for determining the chromatic number of a graph are proved usually to take a number of steps which grows faster than exponentially with the number of vertices in the graph. A similar result holds for the number of steps in certain proofs of lower bounds for chromatic numbers.