We address here the problem of generating random graphs uniformly from the set of simple connected graphs having a prescribed degree sequence. Our goal is to provide an algorithm designed for practical use both because of its ability to generate very large graphs (efficiency) and because it is easy to implement (simplicity). We focus on a family of heuristics for which we prove optimality condi...

The reformulated Zagreb indices of a graph is obtained from the classical Zagreb by replacing vertex degree by edge degree and are defined as sum of squares of the degree of the edges and sum of product of the degrees of the adjacent edges. In this paper we give some explicit results for calculating the first and second reformulated Zagreb indices of dendrimers. Mathematics Subject Classificati...

In 1995, Stiebitz [13] asked the following question: For any positive integers s, t, is there a finite integer f(s, t) such that every digraph D with minimum out-degree at least f(s, t) admits a bipartition (A,B) such that A induces a subdigraph with minimum out-degree at least s and B induces a subdigraph with minimum out-degree at least t? We give an affirmative answer for tournaments, bipart...

Sacks [14] showed that every computably enumerable (c.e.) degree ≥ 0 has a c.e. splitting. Hence, relativising, every c.e. degree has a Δ2 splitting above each proper predecessor (by ‘splitting’ we understand ‘nontrivial splitting’). Arslanov [1] showed that 0′ has a d.c.e. splitting above each c.e. a < 0′. On the other hand, Lachlan [9] proved the existence of a c.e. a > 0 which has no c.e. sp...

The Major Subdegree Problem of A. H. Lachlan (first posed in 1967) has become the longest-standing open question concerning the structure of the computably enumerable (c.e.) degrees. A c.e. degree a is a major subdegree of a c.e. degree b > a if for any c.e. degree x, 0′ = b ∨ x if and only if 0′ = a ∨ x. In this paper, we show that every c.e. degree b 6= 0 or 0′ has a major subdegree, answerin...

In a seminal STOC’95 paper, Arya et al. [4] devised a construction that for any set S of n points in R and any ǫ > 0, provides a (1 + ǫ)-spanner with diameter O(log n), weight O(log n)w(MST (S)), and constant maximum degree. Another construction of [4] provides a (1 + ǫ)-spanner with O(n) edges and diameter α(n), where α stands for the inverse-Ackermann function. Das and Narasimhan [18] devised...

Grone and Merris [5] conjectured that the Laplacian spectrum of a graph is majorized by its conjugate vertex degree sequence. In this paper, we prove that this conjecture holds for a class of graphs including trees. We also show that this conjecture and its generalization to graphs with Dirichlet boundary conditions are equivalent.