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 of Graph Theory – Wiley
Published: Jan 1, 2018
Keywords: ; ; ; ; ; ; ; ; ;
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