Abstract
The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of an n-vertex maximal planar graph is at most lfloor frac{1}{18}(n^3+3n^2)rfloor . We prove this conjecture and determine the unique n-vertex maximal planar graph attaining this maximum, for every nge 10.
Highlights
We prove this conjecture and determine the unique n-vertex maximal planar graph attaining this maximum, for every n ≥ 10
The Wiener index is a graph invariant based on distances in the graph
For a connected graph G, the Wiener index is the sum of distances between all unordered pairs of vertices in the graph and is denoted by W (G)
Summary
The Wiener index is a graph invariant based on distances in the graph. For a connected graph G, the Wiener index is the sum of distances between all unordered pairs of vertices in the graph and is denoted by W (G). Journal of Combinatorial Optimization (2020) 40:1121–1135 where dG (u, v) denotes the distance from u to v i.e. the minimum length of a path from u to v in the graph G It was first introduced by Wiener (1947) while studying its correlations with boiling points of paraffin considering its molecular structure. The Wiener index of a maximal planar graph with n vertices, n ≥ 3 has a sharp lower bound (n − 2)2 + 2. Che and Collins (2018), and independently Czabarka et al (2019), gave a sharp upper bound of a particular class of maximal planar graphs known as Apollonian networks. Equality holds if and only if G is isomorphic to Tn for all n ≥ 9
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: 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.
