Abstract
We consider a subcritical Galton–Watson tree $\mathsf {T}_{n}^{\Omega }$ conditioned on having $n$ vertices with outdegree in a fixed set $\Omega $. The offspring distribution is assumed to have a regularly varying density such that it lies in the domain of attraction of an $\alpha $-stable law for $1<\alpha \le 2$. Our main results consist of a local limit theorem for the maximal degree of $\mathsf {T}_{n}^{\Omega }$, and various limits describing the global shape of $\mathsf {T}_{n}^{\Omega }$. In particular, we describe the joint behaviour of the fringe subtrees dangling from the vertex with maximal degree. We provide applications of our main results to establish limits of graph parameters, such as the height, the non-maximal vertex outdegrees, and fringe subtree statistics.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.