It is well-known that the second smallest eigenvalue 22 of the difference Laplacian matrix of a graph G is related to the expansion properties of G. A more detailed analysis of this relation is given. Upper and lower bounds on the diameter and the mean distance in G in terms of 22 are derived.

Given a set P of n points in a plane, the Geometric Minimum Diameter Spanning Tree (GMDST) of P is a tree that spans P and minimizes the Euclidian length of the longest path. The best known algorithm for this problem runs in slightly sub-cubic time. In this report, I have surveyed some algorithms and approximation schemes developed for this problem and investigated the relation between this pro...

Given a set P of points in the plane, a geometric minimum-diameter spanning tree (GMDST) of P is a spanning tree of P such that the longest path through the tree is minimized. In this paper, we present an approximation algorithm that generates a tree whose diameter is no more than (1+ ) times that of a GMDST, for any > 0. Our algorithm reduces the problem to several grid-aligned versions of the...

Let P be a set of n points in the plane. The geometric minimum-diameter spanning tree (MDST) of P is a tree that spans P and minimizes the Euclidian length of the longest path. It is known that there is always a monoor a dipolar MDST, i.e. a MDST whose longest path consists of two or three edges, respectively. The more difficult dipolar case can so far only be solved in O(n) time. This paper ha...

The diameter of a set P of n points in R is the maximum Euclidean distance between any two points in P . If P is the vertex set of a 3-dimensional convex polytope, and if the combinatorial structure of this polytope is given, we prove that, in the worst case, deciding whether the diameter of P is smaller than 1 requires Ω(n log n) time in the algebraic computation tree model. It shows that the ...

Let n be a positive integer, and λ > 0 a real number. Let Vn be a set of n points randomly located within the unit disk, which are mutually independent. For 1 ≤ p ≤ ∞, define Gp(λ, n) to be the graph with the vertex set Vn, in which two vertices are adjacent if and only if their `p-distance is at most λ. We call this graph a unit disk random graph. Let λ = c √ ln n/n and let X be the number of ...

some physical attributes of two common types of iranian garlic cloves (white and pink) were identified and compared. a machine vision system was used to determine three di-mensions and both major and minor projected areas of garlic cloves at a moisture content of 42.4% w. b. the geometric mean diameter and sphericity were calculated, as well as the unit mass and volume of cloves were measured. ...

The paper discusses the asymptotic behavior of generalizations of the Gauss’s arithmetic-geometric mean, associated with the names Meissel (1875) and Borchardt (1876). The "hapless computer experiment" in the title refers to the fact that the author at an earlier stage thought that one had genuine asymptotic formulae but it is now shown that in general "fluctuations" are present. However, no ve...