On certain regular nicely distance-balanced graphs
Blas Fernandez, \v{S}tefko Miklavi\v{c}, Safet Penji\'c

TL;DR
This paper classifies regular nicely distance-balanced graphs where the parameter b3b3 equals the diameter plus one, extending the understanding of their structure beyond known cases.
Contribution
It provides a complete classification of regular nicely distance-balanced graphs with b3 = d+1, a case not previously characterized.
Findings
Regular nicely distance-balanced graphs with b3 = d+1 are classified.
Known cases include complete graphs and certain cycles for b3 = d.
The paper extends the classification to new graph families.
Abstract
A connected graph is called {\em nicely distance--balanced}, whenever there exists a positive integer , such that for any two adjacent vertices of there are exactly vertices of which are closer to than to , and exactly vertices of which are closer to than to . Let denote the diameter of . It is known that , and that nicely distance-balanced graphs with are precisely complete graphs and cycles of length or . In this paper we classify regular nicely distance-balanced graphs with .
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Taxonomy
TopicsGraph theory and applications · Finite Group Theory Research · Advanced Graph Theory Research
