Dankelmann, P. Size of graphs and digraphs with given diameter and connectivity constraints. (English) Zbl 1488.05153 Acta Math. Hung. 164, No. 1, 178-199 (2021). For a connected, finite graph or a strongly connected finite digraph, the diameter is the largest of all distances between pairs of vertices; the edge-connectivity is the minimum number of edges (respectively arcs) whose removal renders a graph \(G\) (respectively a strong digraph \(G\)) disconnected (respectively not strongly connected); a strongly connected digraph is Eulerian if, for every vertex, the in-degree and out-degree coincide. In the first instance this paper addresses gaps in the literature relating, for undirected \(G=(V,E)\), \(\vert V\vert \), \(\vert E\vert \), diameter, and edge-connectivity. It also seeks, in §4, to generalize results to Eulerian digraphs, relating e.g. P. Dankelmann and L. Volkmann [Electron. J. Comb. 17, No. 1, Research Paper R157, 11 p. (2010; Zbl 1204.05042)]. In Theorem 1 the author determines for graphs \(G=(V,E)\) the maximum of \(\vert E\vert \) given \(\vert V\vert \), diameter, and edge-connectity \(\lambda\), where \(2\le\lambda\le 7\); for \(\lambda=1\) this is a classical result. Reviewer: William G. Brown (Montréal) MSC: 05C12 Distance in graphs 05C20 Directed graphs (digraphs), tournaments Keywords:diameter; connectivity; edge-connectivity; digraph; Eulerian digraph Citations:Zbl 1204.05042 PDFBibTeX XMLCite \textit{P. Dankelmann}, Acta Math. Hung. 164, No. 1, 178--199 (2021; Zbl 1488.05153) Full Text: DOI References: [1] Ali, P.; Mazorodze, JP; Mukwembi, S.; Vetrík, T., On size, order, diameter and edge-connectivity of graphs, Acta Math. Hungar., 152, 11-24 (2017) · Zbl 1399.05117 [2] Caccetta, L.; Smyth, WF, Properties of edge-maximal \(K\)-edge-connected \(D\)-critical graphs, J. Combin. Math. Combin. Comput., 2, 111-131 (1987) · Zbl 0635.05025 [3] Caccetta, L.; Smyth, WF, Redistribution of vertices for maximum edge count in \(K\)-edge-connected \(D\)-critical graphs, Ars Combin., 26, 115-132 (1988) · Zbl 0671.05043 [4] Caccetta, L.; Smyth, WF, Graphs of maximum diameter, Discrete Math., 102, 121-141 (1992) · Zbl 0755.05046 [5] Dankelmann, P.; Volkmann, L., The diameter of almost Eulerian digraphs, Electron. J. Combin., 17, R157 (2010) · Zbl 1204.05042 [6] Dankelmann, P., Distance and size in digraphs, Discrete Math., 338, 144-148 (2015) · Zbl 1301.05097 [7] Dankelmann, P.; Dorfling, M., Diameter and maximum degree in Eulerian digraphs, Discrete Math., 339, 1355-1361 (2016) · Zbl 1329.05126 [8] Erdős, P.; Pach, J.; Pollack, R.; Tuza, Z., Radius, diameter, and minimum degree, J. Combin. Theory Ser. B, 47, 73-79 (1989) · Zbl 0686.05029 [9] Mukwembi, S., A note on diameter and the degree sequence of a graph, Appl. Math. Lett., 25, 175-178 (2012) · Zbl 1234.05072 [10] Mukwembi, S., On size, order, diameter and minimum degree, Indian J. Pure Appl. Math., 44, 467-472 (2013) · Zbl 1277.05054 [11] Ore, O., Diameters in graphs, J. Combin. Theory, 5, 75-81 (1968) · Zbl 0175.20804 [12] N. A. Ostroverhiĭ, V. V. Strok and N. P. Homenko, Diameter critical graphs and digraphs, in: Topological Aspects of Graph Theory, Izdanie Inst. Mat. Akad. Nauk. Ukrain. SSR (Kiev, 1971), pp. 214-271 (in Russian) [13] Plesník, J., On the sum of all distances in a graph or digraph, J. Graph Theory, 8, 1-24 (1984) · Zbl 0552.05048 [14] Rho, Y.; Kim, BM; Hwang, W.; Song, BC, Minimum orders of Eulerian oriented digraphs with given diameter, Acta Math, Sinica, English Series, 30, 1125-1132 (2014) · Zbl 1304.05077 [15] Soares, J., Maximum diameter of regular digraphs, J. Graph Theory, 16, 437-450 (1992) · Zbl 0768.05048 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.