Guantao Chen

Graphs with linearly bounded Ramsey numbers

Abstract A graph G of order n is p -arrangeable if its vertices can be ordered as v 1 , v 2 , ..., v n such that | N L i ( N Ri ( v i ))| ≤ p for each 1 ≤ i ≤ n − 1, where L i = { v 1 , v 2 , ..., v i }, R i = { v i +1 , v i +2 , ..., v n }, and N A ( B ) denotes the neighbors of B which lie in A . We prove that for each p ≥ 1, there is a constant c (depending only on p ) such that the Ramsey number r ( G , G ) ≤ cn for each p -arrangeable graph G of order n . Further we prove that there exists a fixed positive integer p such that all planar graphs are p -arrangeable.

Degree conditions for 2-factors

For any positive integer k, we investigate degree conditions implying that a graph G of order n contains a 2-factor with exactly k components (vertex disjoint cycles). In particular, we prove that for k ≤ (n/4), Ores classical condition for a graph to be hamiltonian (k = 1) implies that the graph contains a 2-factor with exactly k components. We also obtain a sufficient degree condition for a graph to have k vertex disjoint cycles, at least s of which are 3-cycles and the remaining are 4-cycles for any s ≤ k.

Predicting calcium-binding sites in proteins—A graph theory and geometry approach

Identifying calcium‐binding sites in proteins is one of the first steps towards predicting and understanding the role of calcium in biological systems for protein structure and function studies. Due to the complexity and irregularity of calcium‐binding sites, a fast and accurate method for predicting and identifying calcium‐binding protein is needed. Here we report our development of a new fast algorithm (GG) to detect calcium‐binding sites. The GG algorithm uses a graph theory algorithm to find oxygen clusters of the protein and a geometric algorithm to identify the center of these clusters. A cluster of four or more oxygen atoms has a high potential for calcium binding. High performance with about 90% site sensitivity and 80% site selectivity has been obtained for three datasets containing a total of 123 proteins. The results suggest that a sphere of a certain size with four or more oxygen atoms on the surface and without other atoms inside is necessary and sufficient for quickly identifying the majority of the calcium‐binding sites with high accuracy. Our finding opens a new avenue to visualize and analyze calcium‐binding sites in proteins facilitating the prediction of functions from structural genomic information. Proteins 2006.

Transforming Complete Coverage Algorithms to Partial Coverage Algorithms for Wireless Sensor Networks

The complete area coverage problem in Wireless Sensor Networks (WSNs) has been extensively studied in the literature. However, many applications do not require complete coverage all the time. For such applications, one effective method to save energy and prolong network lifetime is to partially cover the area. This method for prolonging network lifetime recently attracts much attention. However, due to the hardness of verifying the coverage ratio, all the existing centralized or distributed but nonparallel algorithms for partial coverage have very high time complexities. In this work, we propose a framework which can transform almost any existing complete coverage algorithm to a partial coverage one with any coverage ratio by running a complete coverage algorithm to find full coverage sets with virtual radii and converting the coverage sets to partial coverage sets via adjusting sensing radii. Our framework can preserve the characteristics of the original algorithms and the conversion process has low time complexity. The framework also guarantees some degree of uniform partial coverage of the monitored area.

An Interlacing Result on Normalized Laplacians

Given a graph G, the normalized Laplacian associated with the graph G, denoted

Towards predicting Ca2+-binding sites with different coordination numbers in proteins with atomic resolution.

{\cal L}

Long cycles in 3-connected graphs

(G), was introduced by F. R. K. Chung and has been intensively studied in the last 10 years. For a k-regular graph G, the normalized Laplacian

A generalization of Fan's condition for Hamiltonicity, pancyclicity, and Hamiltonian connectedness

{\cal L}

Analysis and prediction of calcium‐binding pockets from apo‐protein structures exhibiting calcium‐induced localized conformational changes

(G) and the standard Laplacian matrix L(G) satisfy L(G) = k {\cal L} (G), and hence they have the same eigenvectors and their eigenvalues are directly related. However, for an irregular graph G,

Graph Connectivity After Path Removal

{\cal L} (G) and L(G) behave quite differently, and the normalized Laplacian seems to be more natural. In this paper, Cauchy interlacing-type properties of the normalized Laplacian are investigated, and the following result is established. Let G be a graph, and let H = G - e, where e is an edge of G. Let