Abstract
In this paper, we determine the maximum sizes of strong digraphs under the constraint that their some parameters are fixed, such as vertex connectivity, edge-connectivity, the number of cut vertices. The corresponding extremal digraphs are also characterized. In addition, we establish Nordhaus–Gaddum type theorem for the diameter when \(\overrightarrow{K_n}\) decomposing into many parts. We also pose a related conjecture for Wiener index of digraphs.
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References
An Z, Wu B, Li D, Wang Y, Su G (2011) Nordhaus–Gaddum-type theorem for diameter of graphs when decomposing into many parts. Discret Math Algorithm Appl 3(3):305–310
Balakrishnan R, Sridharan N, Viswanathan Iyer K (2008) Wiener index of graphs with more than one cut-vertex. Appl Math Lett 21:922–927
Bang-Jensen J, Gutin G (2001) Digraphs theory, algorithms and applications. Springer, New York
Bang-Jensen J, Nielsen MH (2008) Minimum cycle factors in quasi-transitive digraphs. Discret Optim 5:121–137
Bereg S, Wang H (2007) Wiener indices of balanced binary trees. Discret Appl Math 155:457–467
Bondy JA, Murty USR (1976) Graph theory with applications. MacMillan, London
Broersma HJ, Li X (2002) Some approaches to a conjecture on short cycles in digraphs. Discret Appl Math 120:45–53
Dankelmann P (2015) Distance and size in digraphs. Discret Math 338:144–148
Dankelmann P, Domke GS, Goddard W, Grobler P, Hattingh JH, Swart HC (2004) Maximum sizes of graphs with given domination parameters. Discret Math 281:137–148
Dobrynin AA, Eentringer R, Gutman I (2001) Wiener index of trees: theory and applications. Acta Appl Math 66:211–249
Eliasi M, Taeri B (2009) Four new sums of graphs and their Wiener indices. Discret Appl Math 157:794–803
Ferneyhough S, Haas R, Hanson D, MacGillivray G (2002) Star forests, dominating sets and Ramsey-type problems. Discret Math 245:255–262
Füredi Z, Kostochka AV, Škrekovski R, Stiebitz M, West DB (2005) Nordhaus–Gaddum-type theorems for decompositions into many parts. J Graph Theory 50:273–292
Goddard W, Henning MA, Swart HC (1992) Some Nordhaus–Gaddum type results. J Graph Theory 16(3):221–231
Huang Z, Lin H (2015) Sizes and transmissions of digraphs with a given clique number. J Comb Optim. doi:10.1007/s10878-015-9850-5
Huang Z, Zhan X (2011) Digraphs that have at most one walk of a given length with the same end points. Discret Math 311:70–79
Li D, Wu B, Yang X, An X (2011) Nordhaus–Gaddum-type theorem for Wiener index of graphs when decomposing into three parts. Discret Appl Math 159:1594–1600
Liu J, Meng J, Zhang Z (2010) Double-super-connected digraphs. Discret Appl Math 158:1012–1016
Nordhaus EA, Gaddum JW (1956) On complementary graphs. Am Math Mon 63:175–177
Sanchis LA (1991) Maximum number of edges in connected graphs with a given domination number. Discret Math 87:65–72
Sanchis LA (2000) On the number of edges in graphs with a given connected domination number. Discret Math 214:193–210
Severini S (2006) On the structure of the adjacency matrix of the line digraph of a regular digraph. Discret Appl Math 154:1763–1765
Vizing VG (1965) A bound on the external stability number of a graph. Dokl Akad Nauk SSSR 164:729–731
Wagner SG, Wang H, Yu G (2009) Molecular graphs and the inverse Wiener index problem. Discret Appl Math 157:1544–1554
Xu S (1991) Relation between diameter of graphs. Discret Math 89(1):65–88
Zhang L, Wu B (2005) The Nordhaus–Gaddum type inequalities of some chemical indices. Commun Math Comput Chem 54:189–194
Acknowledgments
We would like to express our gratitude to the referees for their careful reviews and detailed comments. The first author was supported by the NSFC (Nos. 11401211 and 11471211), the China Postdoctoral Science Foundation (No. 2014M560303) and Fundamental Research Funds for the Central Universities (No. 222201414021). The second author was supported by the NSFC (11471211). The third author was supported by the NSFC (11161046) and by Xingjiang Talent Youth Project (2013721012).
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Lin, H., Shu, J. & Wu, B. Maximum size of digraphs with some parameters. J Comb Optim 32, 941–950 (2016). https://doi.org/10.1007/s10878-015-9916-4
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DOI: https://doi.org/10.1007/s10878-015-9916-4