2021/06/04 by Peter Dankelmann, Sonwabile Mafunda, Dankelmann, Peter +1
Computer Science · Mathematics · #05C12 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph Labeling and Dimension Problems #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.2106.02500
openalex publication_date 2021/06/04 · openalex created_date 2022/07/23 · openalex updated_date 2026/07/28
The average distance of a vertex v of a connected graph G is the arithmetic mean of the distances from v to all other vertices of G. The proximity π(G) and the remoteness ρ(G) of G are the minimum and the maximum of the average distances of the vertices of G. In this paper, we give upper bounds on the difference between the remoteness and proximity, the diameter and proximity, and the radius and proximity of a triangle-free graph with given order and minimum degree. We derive the latter two results by first proving lower bounds on the proximity in terms of order, minimum degree and either diameter or radius. Our bounds are sharp apart from an additive constant. We also obtain corresponding bounds for C4-free graphs.