A simple graph G is representable in a real vector space of dimension m if there is an embedding of the vertex set in the vector space such that the Euclidean distance between any two vertices is one of only two distinct values a or b, with distance a if the vertices are adjacent and distance b otherwise. The Euclidean representation number of G is the smallest dimension in which G is represent...

Graph search is attractive for the quantitative analysis of volumetric medical images, and especially for layered tissues, because it allows globally optimal solutions in low-order polynomial time. However, because nodes of graphs typically encode evenly distributed voxels of the volume with arcs connecting orthogonally sampled voxels in Euclidean space, segmentation cannot achieve greater prec...

Some graphs admit drawings in the Euclidean k-space in such a (natural) way, that edges are represented as line segments of unit length. Such drawings will be called k dimensional unit distance representations. When two non-adjacent vertices are drawn in the same point, we say that the representation is degenerate. The dimension (the Euclidean dimension) of a graph is defined to be the minimum ...

In this paper, we represent protein structure by using graph. A protein structure database will become a graph database. Each graph is represented by a spectral vector. We use Jacobi rotation algorithm to calculate the eigenvalues of the normalized Laplacian representation of adjacency matrix of graph. To measure the similarity between two graphs, we calculate the Euclidean distance between two...

We present polynomial-time approximation schemes for the problem of finding a minimum-cost k-connected Euclidean graph on a finite point set in R. The cost of an edge in such a graph is equal to the Euclidean distance between its endpoints. Our schemes use Steiner points. For every given c > 1 and a set S of n points in R, a randomized version of our scheme finds an Euclidean graph on a superse...

Given a set P of points in the plane, an Euclidean t-spanner for P is a geometric graph that preserves the Euclidean distances between every pair of points in P up to a constant factor t. The weight of a geometric graph refers to the total length of its edges. In this paper we show that the problem of deciding whether there exists an Euclidean t-spanner, for a given set of points in the plane, ...

In 2008, Vallentin made a conjecture involving the least distortion of an embedding distance-regular graph into Euclidean space. Vallentin’s implies that for diameter d, most contracted pairs vertices are those at distance d. this paper, we confirm several families graphs. We also provide counterexamples to conjecture, where largest contraction occurs between d−1. suggest three alternative conj...