We propose an output-sensitive algorithm for the enumeration of all maximal bicliques in a bipartite graph, tailored to the case when the degree distribution in one partite set is very skewed. We accomplish a worst-case bound better than previously known general bounds if, e.g., the degree sequence follows a power law.

A family 9 of distinct k-element subsets of the n-element set X is called intersecting if Fn F’ # @ holds for all F, F’ E 9. Erdos, Ko, and Rado proved that for n > 2k necessarily ) 9 ) < (1: :) holds, which is clearly best possible (take all k-sets through a fixed element). For a family 9, its maximum degree d(9) is the maximum number of sets in .P= containing any particular element of X. For ...

Using a maximum entropy principle to assign a statistical weight to any graph, we introduce a model of random graphs with arbitrary degree distribution in the framework of standard statistical mechanics. We compute the free energy and the distribution of connected components. We determine the size of the percolation cluster above the percolation threshold. The conditional degree distribution on...

Let C be a conjugation class of permutations of a finite field Fq. We consider the function NCðqÞ defined as the number of permutations in C for which the associated permutation polynomial has degree 5q 2. In 1969, Wells proved a formula for N1⁄23 ðqÞ where 1⁄2k denotes the conjugation class of k-cycles. We will prove formulas for N1⁄2k ðqÞ where k 1⁄4 4; 5; 6 and for the classes of permutation...

This paper discusses separable term structure diffusion models in an arbitrage-free environment. Using general consistency results we exploit the interplay between the diffusion coefficients and the functions determining the forward curve. We introduce the particular class of polynomial term structure models. We formulate the appropriate conditions under which the diffusion for a quadratic term...

We describe the structure of those graphs that have largest spectral radius in the class of all connected graphs with a given degree sequence. We show that in such a graph the degree sequence is non-increasing with respect to an ordering of the vertices induced by breadth-first search. For trees the resulting structure is uniquely determined up to isomorphism. We also show that the largest spec...

We give a corrected proof of an extension of the Robinson Splitting Theorem for the d.c.e. degrees. The purpose of this short paper is to clarify and correct the main result and proof contained in [1]. There we gave a simple proof that there exists no low maximal d.c.e. degree. This was obtained as an immediate corollary of the following strengthening of the Robinson Splitting Theorem (Theorem ...

The (∆, D) (degree/diameter) problem consists of finding the largest possible number of vertices n among all the graphs with maximum degree ∆ and diameter D. We consider the (∆, D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We prove that for the (∆, 2) problem, the number of vertices is n = ∆ + 2; and for the (∆, 3) problem, n...

Given a permutation π ∈ Sn, let Γ−(π) be the graph on n vertices {1, . . . , n} where two vertices i < j are adjacent if π(i) > π(j) and there are no integers k, i < k < j, such that π(i) > π(k) > π(j). Let Γ(π) be the graph obtained by dropping the condition that π(i) > π(j), i.e. two vertices are adjacent if the rectangle [i, π(i)] × [j, π(j)] is empty. In the study of the strong order on per...

We study packings of graphs with given maximal degree. We shall prove that the (hitherto unproved) Bollobás–Eldridge–Catlin conjecture holds in a considerably stronger form if one of the graphs is d-degenerate for d not too large: if d,∆1,∆2 ≥ 1 and n > max{40∆1 ln ∆2, 40d∆2} then a d-degenerate graph of maximal degree ∆1 and a graph of order n and maximal degree ∆2 pack. We use this result to ...