## Degree sequences of monocore graphs.(English)Zbl 1295.05192

Summary: A $$k$$-monocore graph is a graph which has its minimum degree and degeneracy both equal to $$k$$. Integer sequences that can be the degree sequence of some $$k$$-monocore graph are characterized as follows: A nonincreasing sequence of integers $$d_{0},\dots, d_{n}$$ is the degree sequence of some $$k$$-monocore graph $$G$$, $$0 \leq k \leq n-1$$, if and only if $$k \leq d_{i} \leq \min \{n - 1, k + n - i\}$$ and $$\sum d_{i} = 2m$$, where $$m$$ satisfies $$\lceil \frac{k\cdot n}{2} \rceil\leq m \leq k \cdot n - {k+1 \choose 2}$$.

### MSC:

 05C75 Structural characterization of families of graphs 05C07 Vertex degrees

### Keywords:

monocore graph; degeneracy; degree sequence

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### References:

 [1] M. Altaf-Ul-Amin, K. Nishikata, T. Koma, T.Miyasato, Y. Shinbo, M. Arifuzzaman, C. Wada, M. Maeda, et al., Prediction of protein functions based on k-cores of protein-protein interaction networks and amino acid sequences, Genome Informatics 14 (2003) 498-499. [2] J. Alvarez-Hamelin, L. Dall’Asta, A. Barrat, A. Vespignani, k-core decomposition: a tool for the visualization of large scale networks, Adv. Neural Inf. Process. Syst. 18 (2006) 41. [3] G. Bader, C. Hogue, An automated method for finding molecular complexes in large protein interaction networks, BMC Bioinformatics 4 (2003). doi:10.1186/1471-2105-4-2 [4] V. Batagelj and M. Zaversnik, An O(m) algorithm for cores decomposition of net- works. http://vlado.fmf.uni-lj.si/pub/networks/doc/cores/cores.pdf. (2002) Last accessed November 25, 2011. · Zbl 1284.05252 [5] A. Bickle, The k-cores of a Graph (Ph.D. Dissertation, Western Michigan University, 2010). [6] A. Bickle, Structural results on maximal k-degenerate graphs, Discuss. Math. Graph Theory 32 (2012) 659-676. doi:10.7151/dmgt.1637 · Zbl 1293.05313 [7] A. Bickle, Cores and shells of graphs, Math. Bohemica 138 (2013) 43-59. · Zbl 1274.05399 [8] B. Bollobas, Extremal Graph Theory (Academic Press, 1978). · Zbl 0419.05031 [9] M. Borowiecki, J. Ivanˇco, P. Mih´ok and G. Semaniˇsin, Sequences realizable by max- imal k-degenerate graphs, J. Graph Theory 19 (1995) 117-124. doi:10.1002/jgt.3190190112 · Zbl 0813.05061 [10] G. Chartrand and L. Lesniak, Graphs and Digraphs, (4th Ed.) (CRC Press, 2005). [11] S. Dorogovtsev, A. Goltsev and J. Mendes, k-core organization of complex networks, Phys. Rev. Lett. 96 (2006). [12] Z. Filáková, P. Mihók and G. Semaniˇsin., A note on maximal k-degenerate graphs, Math. Slovaca 47 (1997) 489-498. · Zbl 0937.05040 [13] M. Gaertler and M. Patrignani, Dynamic analysis of the autonomous system graph, Proc. 2nd International Workshop on Inter-Domain Performance and Simulation (2004) 13-24. [14] D.R. Lick and A.T. White, k-degenerate graphs, Canad. J. Math. 22 (1970) 1082-1096. doi:10.4153/CJM-1970-125-1 · Zbl 0202.23502 [15] T. Luczak, Size and connectivity of the k-core of a random graph, Discrete Math. 91 (1991) 61-68. doi:10.1016/0012-365X(91)90162-U [16] J. Mitchem, Maximal k-degenerate graphs, Util. Math. 11 (1977) 101-106. · Zbl 0348.05109 [17] S.B. Seidman, Network structure and minimum degree, Soc. Networks 5 (1983) 269-287. doi:10.1016/0378-8733(83)90028-X [18] J.M.S. Simões-Pereira, A survey of k-degenerate graphs, Graph Theory Newsletter 5 (1976) 1-7. · Zbl 0331.05135 [19] D. West, Introduction to Graph Theory, (2nd Ed.) (Prentice Hall, 2001). [20] S. Wuchty and E. Almaas, Peeling the yeast protein network, Proteomics 5 (2005) 444-449. doi:10.1002/pmic.200400962
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