By Aitken R.J.

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These investigations finally lead to the following theorem which generalizes the central limit theorem for cartesian products of graphs. V; E/ be a finite connected graph with jVj 2. For N 1 and k 1 let GŒN;k be the distance-k graph of N G DG G (N-fold Cartesian power) and AŒN;k its adjacency matrix. Then, for a fixed k 1, the eigenvalue distribution of N k=2 AŒN;k converges in moments as N ! 0; 1/. In this note we study the distribution (with respect to the vacuum state) of the distance-k graph of the star product of graphs.

J. Aldous, The continuum random tree. II. An overview, in Stochastic Analysis (Durham, 1990). Volume 167 of London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, 1991), pp. 23–70. edu/~aldous/Papers/me55. pdf 5. J. M. Steele, The objective method: probabilistic combinatorial optimization and local weak convergence. Probab. Discret. Struct. 110, 1–72 (2004). berkeley. pdf 6. E. Baur, On a ternary coalescent process. ALEA Lat. Am. J. Probab. Math. Stat. 10(2), 561–589 (2013).

9 Discrete Coalescents 23 weights, and let H be the subgraph of Kn with the same vertices, but containing only edges of weight at most 1=n. A tree component of H is a connected component of H that is a tree. Exercise 17 (a) Let N be the number of vertices in tree components of H of size at most bn1=4 c. Using Chebyshev’s inequality, show that P fN D 0g ! 0 as n ! 1. (b) Fix S f1; : : : ; ng. Show that, given that H contains a tree component whose vertices are precisely S, then such a component is uniformly distributed over labeled trees with vertices S.