Search

In 1958 Carleson proved that a sequence is interpolating if it is uniformly separated, later in 1983 Peter Jones gave a simple constructive proof in [JON] which is stunning in his time. On the other hand, when a sequence is not uniformly separated, Garnett gave a sharp positive result in [GAR2]. Following Jones's idea in [JON] and also a highly relevant paper [VIN] due to Vinogradov, we give a constructive proof to Garnett's result on nonuniformly separated sequence with slightly modified assumption. We will also discuss extremal problems that is described in the language of Hardy spaces and solve it in functional analytical point of view.

In 1958 Carleson proved that a sequence is interpolating if it is uniformly separated, later in 1983 Peter Jones gave a simple constructive proof in [JON] which is stunning in his time. On the other hand, when a sequence is not uniformly separated, Garnett gave a sharp positive result in [GAR2]. Following Jones's idea in [JON] and also a highly relevant paper [VIN] due to Vinogradov, we give a constructive proof to Garnett's result on nonuniformly separated sequence with slightly modified assumption. We will also discuss extremal problems that is described in the language of Hardy spaces and solve it in functional analytical point of view.

In 1958 Carleson proved that a sequence is interpolating if it is uniformly separated, later in 1983 Peter Jones gave a simple constructive proof in [JON] which is stunning in his time. On the other hand, when a sequence is not uniformly separated, Garnett gave a sharp positive result in [GAR2]. Following Jones's idea in [JON] and also a highly relevant paper [VIN] due to Vinogradov, we give a constructive proof to Garnett's result on nonuniformly separated sequence with slightly modified assumption. We will also discuss extremal problems that is described in the language of Hardy spaces and solve it in functional analytical point of view.

The Erdős-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k− 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ζ(G) of a graph G defined as ζ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ζ(G) ≥ 2k − 4 contains every tree of k edges if d G (x) + d G (y) ≥ 2k − 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d G (u) ≥ k., Postprint (published version)

A well-known fundamental problem in extremal graph theory is the degree/diameter problem, which is to determine the largest (in terms of the number of vertices) graphs or digraphs or mixed graphs of given maximum degree, respectively, maximum outdegree, respectively, mixed degree; and given diameter. General upper bounds, called Moore bounds, exist for the largest possible order of such graphs, digraphs and mixed graphs of given maximum degree d (respectively, maximum out-degree d, respectively, maximum mixed degree) and diameter k. In recent years, there have been many interesting new results in all these three versions of the problem, resulting in improvements in both the lower bounds and the upper bounds on the largest possible number of vertices. However, quite a number of questions regarding the degree/diameter problem are still wide open. In this paper we present an overview of the current state of the degree/diameter problem, for undirected, directed and mixed graphs, and we outline several related open problems., Peer Reviewed

Extremal graphs which are close related to Moore graphs have been defined in different ways. Radially Moore graphs are one of these examples of extremal graphs. Although it is proved that radially Moore graphs exist for radius two, the general problem remains open. Knor, and independently Exoo, gives some constructions of these extremal graphs for radius three and small degrees. As far as we know, some few examples have been found for other small values of the degree and the radius. Here, we consider the existence problem of radially Moore graphs of radius three. We use the generalized undirected de Bruijn graphs to give a general construction of radially Moore graphs of radius three and large odd degree., Peer Reviewed