one of the most fundamental features of digital image and the basic steps in image processing, analysis, pattern recognition and computer vision is the edge of an image where the preciseness and reliability of its results will affect directly on the comprehension machine system made objective world. several edge detectors have been developed in the past decades, although no single edge detector...

Chvátal’s Condition is a sufficient condition for a spanning cycle in an n-vertex graph. The condition is that when the vertex degrees are d1, . . . , dn in nondecreasing order, i < n/2 implies that di > i or dn−i ≥ n − i. We prove that this condition cannot hold in both a graph and its complement, and we raise the problem of finding its asymptotic probability in the random graph with edge prob...

We give two results for multicommodity flows in the d-dimensional hypercube Qd with independent random edge-capacities distributed like a random variable C where P[C > 0] > 1/2. Firstly, with high probability as d → ∞, the network can support simultaneous multicommodity flows of volume close to E[C] between all antipodal vertex pairs. Secondly, with high probability, the network can support sim...

W e consider random graphs with edge probability Pn-Q, where n is the number of vertices of the graph, /3 > 0 is fixed, and a = 1 or a = (I + 1)/1 for some fixed positive integer I . We prove that for every firstorder sentence, the probability that the sentence is true for the random graph has an asymptotic limit. Also, there is an eflective procedure for generating the value of the limit in cl...

We give two results for multicommodity flows in the d-dimensional hypercube Qd with independent random edge capacities distributed like C where P[C > 0] > 1/2. Firstly, with high probability as d → ∞, the network can support simultaneous multicommodity flows of volume close to E[C] between all antipodal vertex pairs. Secondly, with high probability, the network can support simultaneous multicom...

Let f be an edge ordering of Kn: a bijection E(Kn) → {1, 2, . . . , ( n 2 ) }. For an edge e ∈ E(Kn), we call f(e) the label of e. An increasing path in Kn is a simple path (visiting each vertex at most once) such that the label on each edge is greater than the label on the previous edge. We let S(f) be the number of edges in the longest increasing path. Chvátal and Komlós raised the question o...

Vertex betweenness centrality is essential in the analysis of social and information networks, and it quantify vertex importance in terms of its quantity of information along geodesic paths in network. Edge betweenness is similar to the vertex betweenness. Co-betweenness centrality is a natural developed notion to extend vertex betweenness centrality to sets of vertices, and pairwise co-between...