Abstract

. The minimal weight of a spanning tree in a complete graph Kn with independent, uniformly distributed random weights on the edges, is shown to have an asymptotic normal distribution. The proof uses a functional limit extension of results by Barbour and Pittel on the distribution of the number of tree components of given sizes in a random graph. 1. Introduction and results Assign random weights T ij , 1 i ! j n, to the edges of the complete graph K n with vertex set f1; : : : ; ng, and let W n be the minimum weight of a spanning tree of K n . We assume that the weights are independent and identically distributed, with a uniform distribution on [0; 1]. It was proved by Frieze [5] that W n ! i(3) = 1 X k=1 k \Gamma3 = 1:202 : : : in probability as n ! 1, see also Bollobs [3]. The main purpose of the present paper is to show that W n has an asymptotic normal distribution. Theorem 1. Let W n be the weight of the minimal spanning tree. Then n 1=2 \Gamma W n \Gamma i(3) \Delta ...

Citations

2262	Convergence of Probability Measures - Billingsley - 1968
1493	Random Graphs - Bollobas - 2001
98	A randomized lineartime algorithm to find minimum spanning trees - Karger, Klein, et al. - 1995
81	A theorem on trees - Cayley
31	On tree census and the giant component of sparse random graphs, Random Structures and Algorithms - Pittel - 1990
15	Orthogonal Decompositions and Functional Limit Theorems for Random Graph Statistics - Janson - 1994
12	Multicyclic components in a random graph process, Random Structures and Algorithms 4 - Janson - 1993
8	Poisson convergence and random graphs - Barbour - 1982
7	Some remarks on the theory of trees - R'enyi - 1959
6	The expected number of components under a random mapping function - Kruskal - 1954
2	On the value of a minimal spanning tree problem - Frieze - 1985