Let G be a graph of su¢ ciently large order n; and let the largest eigenvalue (G) of its adjacency matrix satis…es (G) > p bn2=4c: Then G contains a cycle of length t for every t n=320: This condition is sharp: the complete bipartite graph T2 (n) with parts of size bn=2c and dn=2e contains no odd cycles and its largest eigenvalue is equal to p bn2=4c: This condition is stable: if (G) is close t...

Introduction Managing, analyzing and extracting information from multidimensional data are very important activities in a broad range of domains, like business planning and integration, scientific research, statistics and even distributed systems. In this paper I consider problems like the largest empty circle and hyper-rectangle in a given set of points, deciding containment in sets of circles...

Changes in Sertoli cell numbers and testicular structure during normal development and compensatory hypertrophy were assessed in crossbred Meishan x White Composite males. Boars were assigned at birth to unilateral castration at 1, 10, 56, or 112 days or to remain as intact controls through 220 days. The first testes removed were compared to assess testicular development. At 220 days, testicula...

A maximal independent set is an independent set that is not a proper subset of any other independent set. Liu [J.Q. Liu, Maximal independent sets of bipartite graphs, J. Graph Theory, 17 (4) (1993) 495-507] determined the largest number of maximal independent sets among all n-vertex bipartite graphs. The corresponding extremal graphs are forests. It is natural and interesting for us to consider...

Suppose that d > 1 is the largest power of two that divides the order of a finite quasigroup Q. It then follows that each automorphism of Q must contain a cycle of length not divisible by d in its disjoint cycle decomposition. The proof is obtained by considering the action induced by the automorphism on a certain orientable surface originally described in a more restricted context by Norton an...

The cyclicity of a graph is the largest integer n for which the graph is contractible to the cycle on n vertices. By analyzing the cycle space of a graph, we establish upper and lower bounds on cyclicity. These bounds facilitate the computation of cyclicity for several classes of graphs, including chordal graphs, complete n-partite graphs, n-cubes, products of trees and cycles, and planar graph...

Given a two-dimensional space, let S be a set of points stored in an R-tree, let R be the minimum rectangle containing the elements of S, and let q be a query point such that q / ∈ S and R ∩ q 6= ∅. In this paper, we present an algorithm for finding the empty rectangle with the largest area, sides parallel to the axes of the space, and containing only the query point q. The idea behind algorith...

Let P be a set of n points in an axis-parallel rectangle B in the plane. We present an O(nα(n) log n)-time algorithm to preprocess P into a data structure of size O(nα(n) log n), such that, given a query point q, we can find, in O(log n) time, the largest-area axis-parallel rectangle that is contained in B, contains q, and its interior contains no point of P . This is a significant improvement ...

A strongly connected digraph D of order n is primitive (aperiodic) provided the greatest common divisor of its directed cycle lengths equals 1. For such a digraph there is a minimum integer t, called the exponent of D, such that given any ordered pair of vertices x and y there is a directed walk from x to y of length t. The exponent of D is the largest of n ‘generalized exponents’ that may be a...