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Abstract
We determine the exact value of the Wiener index, the edgeWiener index, and the vertexedgeWiener index of the Basilica graphs, i.e., the sequence of finite Schreier graphs associated with the action of the Basilica group on the rooted binary tree. Moreover, we give a formula for the total distance of every vertex in the Basilica graphs, and we are able to make it explicit for some special vertices. We finally introduce the notions of asymptotic Wiener index and asymptotic total distance, which are compatible with that of convergence of the sequence of finite Basilica graphs to an infinite orbital limit graph in the Gromov–Hausdorff topology: the asymptotic values are explicitly computed.
Original language  English 

Pages (fromto)  3249 
Number of pages  18 
Journal  Discrete Applied Mathematics 
Volume  307 
DOIs  
Publication status  Published  30 Jan 2022 
Keywords
 Basilica graphs
 EdgeWiener index
 Gromov–Hausdorff topology
 Total distance
 VertexedgeWiener index
 Wiener index
ASJC Scopus subject areas
 Applied Mathematics
 Discrete Mathematics and Combinatorics
Fields of Expertise
 Information, Communication & Computing
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 1 Active

Doctoral Program: Discrete Mathematics
Ebner, O., Lehner, F., Greinecker, F., Burkard, R., Wallner, J., Elsholtz, C., Woess, W., Raseta, M., Bazarova, A., Krenn, D., Lehner, F., Kang, M., Tichy, R., SavaHuss, E., Klinz, B., Heuberger, C., Grabner, P., Barroero, F., Cuno, J., Kreso, D. & Berkes, I.
1/05/10 → 31/12/22
Project: Research project