The mean order of sub‐k‐trees of k‐trees

The mean order of sub‐k‐trees of k‐trees This article focuses on the problem of determining the mean orders of sub‐k‐trees of k‐trees. It is shown that the problem of finding the mean order of all sub‐k‐trees containing a given k‐clique C, can be reduced to the previously studied problem of finding the mean order of subtrees of a tree that contain a given vertex. This problem is extended in two ways. The first of these extensions focuses on the mean order of sub‐k‐trees containing a given sub‐k‐tree. The second extension focuses on the expected number of r‐cliques, 1≤r≤k+1, in a randomly chosen sub‐k‐tree containing a fixed sub‐k‐tree X. Sharp lower bounds for both invariants are derived. The article concludes with a study of global mean orders of sub‐k‐trees of a k‐tree. For a k‐tree, from the class of simple‐clique k‐trees, it is shown that the mean order of its sub‐k‐trees is asymptotically equal to the mean subtree order of its dual. For general k‐trees a recursive generating function for the number of sub‐k‐trees of a given k‐tree T is derived. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Graph Theory Wiley

The mean order of sub‐k‐trees of k‐trees

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Publisher
Wiley Subscription Services, Inc., A Wiley Company
Copyright
Copyright © 2018 Wiley Periodicals, Inc.
ISSN
0364-9024
eISSN
1097-0118
D.O.I.
10.1002/jgt.22185
Publisher site
See Article on Publisher Site

Abstract

This article focuses on the problem of determining the mean orders of sub‐k‐trees of k‐trees. It is shown that the problem of finding the mean order of all sub‐k‐trees containing a given k‐clique C, can be reduced to the previously studied problem of finding the mean order of subtrees of a tree that contain a given vertex. This problem is extended in two ways. The first of these extensions focuses on the mean order of sub‐k‐trees containing a given sub‐k‐tree. The second extension focuses on the expected number of r‐cliques, 1≤r≤k+1, in a randomly chosen sub‐k‐tree containing a fixed sub‐k‐tree X. Sharp lower bounds for both invariants are derived. The article concludes with a study of global mean orders of sub‐k‐trees of a k‐tree. For a k‐tree, from the class of simple‐clique k‐trees, it is shown that the mean order of its sub‐k‐trees is asymptotically equal to the mean subtree order of its dual. For general k‐trees a recursive generating function for the number of sub‐k‐trees of a given k‐tree T is derived.

Journal

Journal of Graph TheoryWiley

Published: Jan 1, 2018

Keywords: ; ; ; ; ; ; ; ; ;

References

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