Krishnamoorthy, V.; Thulasiraman, K.; Swamy, M. N. S. Minimum order graphs with specified diameter, connectivity, and regularity. (English) Zbl 0672.05048 Networks 19, No. 1, 25-46 (1989). Summary: Relationships among graph invariants such as the number of vertices, diameter, connectivity, maximum and minimum degrees, and regularity are being studied recently, motivated by their usefulness in the design of fault-tolerant and low-cost communication and interconnection neworks. A graph is called a (d,c,r) graph if it has diameter d, connectivity c, and regularity r. The minimum number of vertices in (d,1,3), (d,2,3), (d,3,3), and (d,c,c) graphs have been reported in the literature. In this paper, the minimum number of vertices in a (d,c,r) graph with \(r>c\) is determined, thereby exhausting all the possible choices of values for d,c, and r. Our proof is constructive and hence we get a collection of optimal (d,c,r) graphs. Cited in 2 Documents MSC: 05C35 Extremal problems in graph theory 05C38 Paths and cycles 05C40 Connectivity 94C15 Applications of graph theory to circuits and networks Keywords:graph invariants; (d,c,r) graph; diameter; connectivity; regularity; minimum number of vertices PDFBibTeX XMLCite \textit{V. Krishnamoorthy} et al., Networks 19, No. 1, 25--46 (1989; Zbl 0672.05048) Full Text: DOI References: [1] Bhattacharya, IEEE Trans. Circuits and Systems 32 pp 407– (1985) [2] Boesh, J. Graph Theory 4 pp 363– (1980) [3] Boesch, Dis. Appl. Math. 3 pp 9– (1981) [4] Boesch, Networks 12 pp 341– (1982) [5] Boesch, J. Graph Theory 8 pp 487– (1984) [6] Boesch, IEEE Trans. Circuits and Systems 32 pp 1289– (1985) [7] Esfahanian, J. Graph Theory 9 pp 503– (1985) [8] Harary, Proc. Natl. Acad. Sci., U.S.A. 48 pp 1142– (1962) [9] He, Networks 14 pp 337– (1984) [10] Imase, IEEE Trans. Comput. 34 pp 267– (1985) [11] Klee, J. Comb. Theory-B 28 pp 184– (1980) [12] Klee, Math. Oper. Res. 1 pp 28– (1976) [13] Klee, J. Comb. Theory-B 23 pp 83– (1977) [14] Myers, IEEE Trans. Circuits and Systems 27 pp 214– (1980) [15] Myers, IEEE Trans. Circuits and Systems 27 pp 698– (1980) [16] Myers, Discre. Maths. 33 pp 289– (1981) [17] Schumacher, Networks 14 pp 63– (1984) [18] and , Graphs, Networks, and Algorithms, Wiley-Interscience, New York, 1981. 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.