We study the Narayana triangles and related families of polynomials. We link this study to Riordan arrays and Hankel transforms arising from a special case of capacity calculation related to MIMO communication systems. A link is established between a channel capacity calculation and a series reversion.

We develop polynomials in z ∈ C for which some generalized harmonic numbers are special cases at z= 0. By using the Riordan array method, we explore interesting relationships between these polynomials, the generalized Stirling polynomials, the Bernoulli polynomials, the Cauchy polynomials and the Nörlund polynomials. © 2007 Elsevier B.V. All rights reserved.

We present a simple proof of the result published by V. Karakhanyan [4] and O. Riordan [6] concerning the vertex-isoperimetric problem on the n-dimensional torus. The proof method is a reduction of this problem to a similar problem on some (2n)-dimensional grid, for which a solution is known [3]. We also simplify and clarify the structure of the involved isoperimetric order.

We give formulas for the Tutte polynomials of parallel and series connections of weightedmatroids, generalizing and unifying formulas of Oxley and Welsh (Discrete Math. 109 (1992) 185), Bollobás and Riordan (Comb. Probab. Comput. 8 (1999) 45) and Woodall (Discrete Math. 247 (2002) 201). © 2004 Elsevier B.V. All rights reserved.

We introduce a new model of random multigraphs with colored vertices and weighted edges. It is similar to the inhomogeneous random graph model of Söderberg [18], extended by Bollobás, Janson and Riordan [6]. By means of analytic combinatorics, we then analyze the birth of complex components, which are components with at least two cycles. We apply those results to give a complete picture of the ...

Bollobás, Brightwell and Leader [2] showed that there are at most 2( n 2)+o(n 2) 2-SAT functions on n variables, and conjectured that in fact almost every 2-SAT function is unate: i.e., has a 2-SAT formula in which no variable’s positive and negative literals both appear. We prove their conjecture, finding the number of 2-SAT functions on n variables to be 2( n 2)+n(1 + o(1)). As a corollary of...

A bipartition of the vertex set of a graph is called balanced if the sizes of the sets in the bipartition differ by at most one. Bollobás and Scott [3] conjectured that if G is a graph with minimum degree at least 2 then V (G) admits a balanced bipartition V1, V2 such that for each i, G has at most |E(G)|/3 edges with both ends in Vi. The minimum degree condition is necessary, and a result of B...

There are various algorithms known for generating graphs with n vertices of given degrees uniformly at random. Unfortunately, none of them is of practical use for all degree sequences, even for those with all degrees equal. In this paper we examine an algorithm which, although it does not generate uniformly at random, is provably close to a uniform generator when the degrees are relatively smal...

A conjecture by Bollobás and Komlós states the following: For every γ > 0 and integers r ≥ 2 and ∆, there exists β > 0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least ( r−1 r + γ)n and H is an r-chromatic graph with n vertices, bandwidth at most βn and maximum degree at most ∆, then G contains a copy of H. This conjecture generalises s...