On ordinary and reverse Wiener indices of non-caterpillars

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Abstract

A tree is known as a caterpillar if the removal of all pendant vertices makes it a path. Otherwise, it is a non-caterpillar. From among all n-vertex non-caterpillars with given diameter d, we find the unique tree formed by attaching the path P2 and nd3 pendant vertices to a center of the path on d+1 vertices with minimum Wiener index, where 4dn3, and we determine the n-vertex non-caterpillars with the kth greatest reverse Wiener indices for all k up to n32 if 8n26 and up to n32a(n)+1 or n32a(n)+2 depending on whether a(n)=n32n2+2n+28 is an integer or not for even n, and n32b(n)+1 or n32b(n)+2 depending on whether b(n)=n32n2+2n+58 is an integer or not for odd n if n27.

Keywords

Wiener index
Reverse Wiener index
Caterpillar
Tree
Diameter

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