Ai, Jiangdong; Gerke, Stefanie; Gutin, Gregory; Mafunda, Sonwabile Proximity and remoteness in directed and undirected graphs. (English) Zbl 1467.05056 Discrete Math. 344, No. 3, Article ID 112252, 7 p. (2021). For a vertex \(v\) of a strongly connected digraph \(D\), the average distance \(\overline{\sigma}(v)\) is the arithmetic mean of the distances from \(v\) to all other vertices of \(D\). The remoteness \(\rho(D)\) and proximity \(\pi(D)\) of \(D\) are the maximum and the minimum of the average distances of the vertices of \(D\), respectively. Both parameters were introduced by B. Zelinka [Arch. Math., Brno 4, 87–95 (1968; Zbl 0206.26105)] and, independently, by M. Aouchiche and P. Hansen [Networks 58, No. 2, 95–102 (2011; Zbl 1232.05062)] for undirected graphs. In the paper under review, the authors extend their study to directed graph, and obtain sharp upper and lower bounds on both parameters in terms of the order of a digraph. As a special case, tournaments are considered. It is shown that for a strong tournament remoteness and proximity coincide if and only if the tournament is regular. In addition, an infinite family of non-regular strong digraphs \(D\) such that \(\rho(D)=\pi(D)\) is presented. Reviewer: Aleksandra Tepeh (Duplek) Cited in 4 Documents MSC: 05C12 Distance in graphs 05C20 Directed graphs (digraphs), tournaments 05C40 Connectivity Keywords:average distance; strongly connected oriented graph; tournament; regular graph; regular digraph Citations:Zbl 0206.26105; Zbl 1232.05062 PDFBibTeX XMLCite \textit{J. Ai} et al., Discrete Math. 344, No. 3, Article ID 112252, 7 p. (2021; Zbl 1467.05056) Full Text: DOI arXiv References: [1] Aouchiche, M.; Hansen, P., Proximity and remoteness in graphs: Results and conjectures, Networks, 58, 2, 95-102 (2011) · Zbl 1232.05062 [2] Bang-Jensen, J.; Gutin, G., Digraphs: Theory, Algorithms and Applications (2009), Springer: Springer London · Zbl 1170.05002 [3] Bang-Jensen, J.; Gutin, G., Basic terminology notation results, (Bang-Jensen, J.; Gutin, G., Classes of Directed Graphs (2018), Springer: Springer Cham), Chapter 1 · Zbl 1407.05101 [4] Barefoot, C. A.; Entringer, R. C.; Szekely, L. A., Extremal values for ratios of distances in trees, Discrete Appl. Math., 80, 37-56 (1997) · Zbl 0886.05059 [5] Dankelmann, P., Proximity, remoteness, and minimum degree, Discrete Appl. Math., 184, 223-228 (2015) · Zbl 1311.05048 [6] Dankelmann, P., New bounds on proximity and remoteness in graphs, Commun. Combin. Optim., 1, 1, 28-40 (2016) · Zbl 1365.05073 [7] P. Dankelmann, E. Jonck, S. Mafunda, Proximity and remoteness in triangle-free and \(C_4\)-free graphs in terms of order and minimum degree, Manuscript. [8] Gutin, G., The radii of \(n\)-partite tournaments, Math. Notes, 40, 743-744 (1986) · Zbl 0691.05016 [9] Landau, H. G., On dominance relations and the structure of animal societies 3. The condition for a score structure, Bull. Math. Biophys., 15, 143-148 (1953) [10] Zelinka, B., Medians and peripherians of trees, Arch. Math., 4, 87-95 (1968) · Zbl 0206.26105 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.