Coalescence and Minimal Spanning Trees of Irregular Graphs

This paper concentrates on breaching the gap between the Smoluchowski coagulation equations for Marcus-Lushnikov processes and the theory of random graphs. It is known that in many cases the cluster dynamics of a random graph process can be replicated with the corresponding coalescent process. The cluster dynamics of a coalescent process (without merger history) is reflected in a auxiliary process called the Marcus-Lushnikov process. The merger dynamics of the Marcus-Lushnikov processes will correspond to a greedy algorithm for finding the minimal spanning tree in the respective random graph process. This observation allows one to express the limiting mean length of a minimal spanning tree in terms of the solutions of the Smoluchowski coagulation equations that represent the hydrodynamic limit of the Marcus-Lushnikov process corresponding to the random graph process. We concentrate on finding the limiting mean length of a minimal spanning tree on an irregular graph. Specifically, an Erdős-Rényi random graph process on the bipartite graph Kα[n],β[n] is considered with α[n] = αn + o(n) and β[n] = βn + o(n). There, the following expression for the limiting mean length of the minimal spanning tree is derived via the Smoluchowski coagulation equations of the Marcus-Lushnikov processes with multidimensional weight vectors: lim n→∞ E[Ln] = γ + 1 γ + ∑ i1≥1; i2≥1 (i1 + i2 − 1)! i1!i2! γi1ii2−1 1 i i1−1 2 (i1 + γi2)12 , where γ = αβ . This is a completely new formula for the case of an irregular bipartite graph γ 6= 1. In the case of γ = 1, the above series adds up to lim n→∞ E[Ln] = 2ζ(3) as derived in Frieze and McDiarmid [14] for a regular bipartite graph. A generalization of the approach is considered in the discussion section.