Ordered trees with prescribed root degrees, node degrees, and branch lengths

Graphs with degrees from prescribed intervals

Constructing Elliptic Curves with Prescribed Embedding Degrees

Pairing-based cryptosystems depend on the existence of groups where the Decision Diffie-Hellman problem is easy to solve, but the Computational Diffie-Hellman problem is hard. Such is the case of elliptic curve groups whose embedding degree is large enough to maintain a good security level, but small enough for arithmetic operations to be feasible. However, the embedding degree for most ellipti...

Degrees in $k$-minimal label random recursive trees

This article describes the limiting distribution of the degrees of nodes has been derived for a kind of random tree named k-minimal label random recursive tree, as the size of the tree goes to infinity. The outdegree of the tree is equal to the number of customers in a pyramid marketing agency immediatly alluring

Spanning Trees with Vertices Having Large Degrees

Let G be a connected simple graph, and let f be a mapping from V (G) to the set of integers. This paper is concerned with the existence of a spanning tree in which each vertex v has degree at least f(v). We show that if ∣∣ΓG(S)∣∣− f(S)+ |S| ≥ 1 for any nonempty subset S ⊆ L, then a connected graph G has a spanning tree such that dT (x) ≥ f(x) for all x ∈ V (G), where ΓG(S) is the set of neighbo...

Spanning trees with minimum weighted degrees

Given a metric graph G, we are concerned with finding a spanning tree of G where the maximum weighted degree of its vertices is minimum. In a metric graph (or its spanning tree), the weighted degree of a vertex is defined as the sum of the weights of its incident edges. In this paper, we propose a 4.5-approximation algorithm for this problem. We also prove it is NP-hard to approximate this prob...