Department of Computing and Information Technology, Fudan University, Shanghai, China 2 Theoretical Systems Biology Lab , School of Life Science, Fudan University, Shanghai, China Human Genetics Center, University of Texas Health Science Center at Houston, Houston, TX 77225, USA Abstract In recent years, evaluating graph distance has become more and more important in a variety of real applicati...

For a given graph G, the square of G, denoted by G2, is a graph with the vertex set V(G) such that two vertices are adjacent if and only if the distance of these vertices in G is at most two. A graph G is called squared if there exists some graph H such that G= H2. A function f:V(G) {0,1,2…, k} is called a coloring of G if for every pair of vertices x,yV(G) with d(x,y)=1 we have |f(x)-f(y)|2 an...

The vertices of the odd-distance graph are the points of the plane R. Two points are connected by an edge if their Euclidean distance is an odd integer. We prove that the chromatic number of this graph is at least five. We also prove that the odd-distance graph in R is countably choosable, while such a graph in R is not.

In this paper, we introduce distance-regular graphs and develop the intersection algebra for these graphs which is based upon its intersection numbers. We discuss results following from the definition of the intersection algebra. We investigate two examples of distance-regular graphs and show how these results apply. Finally, we introduce parameters that determine intersection numbers. We inves...

We study the problem of reconstructing a hidden graph given access to a distance oracle. We design randomized algorithms for the following problems: reconstruction of a degree bounded graph with query complexity Õ(n); reconstruction of a degree bounded outerplanar graph with query complexity Õ(n); and near-optimal approximate reconstruction of a general graph.