From Erdös to algorithms

Fast online graph clustering via Erdös-Rényi mixture

In the context of graph clustering, we consider the problem of estimating simultaneously both the partition of the graph nodes and the parameters of an underlying mixture of affiliation networks. In numerous applications the rapid increase of data size with time makes classical clustering algorithms too slow because of the high computational cost. In such situations online clustering algorithms...

گزارش زنبورپارازیتویید (Microterys hortulanus Erdös ( Hym.: Encyrtidae از ایران

AN EXACT PERFORMANCE BOUND FOR AN O(m+ n) TIME GREEDY MATCHING PROCEDURE

We prove an exact lower bound on γ(G), the size of the smallest matching that a certain O(m+ n) time greedy matching procedure may find for a given graph G with n vertices and m edges. The bound is precisely Erdös and Gallai’s extremal function that gives the size of the smallest maximum matching, over all graphs with n vertices and m edges. Thus the greedy procedure is optimal in the sense tha...

The one - hop neighborhood of Paul Erdös

The idea of the Erdös number was created by his fellow mathematicians as a humorous tribute to his enormous output as one of the most prolific modern writers of mathematical papers [1]. An Erdös number 1 is awarded to the person who has published at least one mathematical paper with the celebrated mathematician. Similarly, joint publications with someone with an Erdös number of 1 yield an Erdös...

The rank of random graphs

We show that almost surely the rank of the adjacency matrix of the Erdös-Rényi random graph G(n, p) equals the number of non-isolated vertices for any c lnn/n < p < 1/2, where c is an arbitrary positive constant larger than 1/2. In particular the giant component (a.s.) has full rank in this range.