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Table of the largest known graphs of a given diameter and maximal degree

Table of the largest known graphs of a given diameter and maximal degree is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Table of the largest known graphs of a given diameter and maximal degree rather than just read about it. In short: In graph theory, the degree diameter problem is the problem of finding the largest possible graph for a given maximum degree and diameter. The Moore bound sets limits on this, but for many years mathematicians in the field have been interested in a more precise answer.

Key takeaways

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Reference excerpt

In graph theory, the degree diameter problem is the problem of finding the largest possible graph for a given maximum degree and diameter. The Moore bound sets limits on this, but for many years mathematicians in the field have been interested in a more precise answer. The table below gives current progress on this problem (excluding the case of degree 2, where the largest graphs are cycles with an odd number of vertices).

Table of the orders of the largest known graphs for the undirected degree diameter problem Below is the table of the vertex numbers for the best-known graphs (as of June 2024) in the undirected degree diameter problem for graphs of degree at most 3 ≤ d ≤ 16 and diameter 2 ≤ k ≤ 10. Only a few of the graphs in this table (marked in bold) are known to be optimal (that is, largest possible). The remainder are merely the largest so far discovered, and thus finding a larger graph that is closer in order (in terms of the size of the vertex set) to the Moore bound is considered an open problem. Some general constructions are known for values of d and k outside the range shown in the table.

Entries without a footnote were found by Loz & Širáň (2008). In all other cases, the footnotes in the table indicate the origin of the graph that achieves the given number of vertices:

References Abas, Marcel (2016), "Cayley graphs of diameter two with order greater than 0.684 of the Moore bound for any degree", European Journal of Combinatorics, 57: 109–120, arXiv:1511.03706, doi:10.1016/j.ejc.2016.04.008 Alegre, Ignacio; Fiol, Miquel; Yebra, J. Luis A. (1986), "Some Large Graphs with Given Degree and Diameter", Journal of Graph Theory, 10 (2): 219–224, doi:10.1002/jgt.3190100211 Allwright, James (1992), "New (Δ, D) graphs discovered by heuristic search", Discrete Applied Mathematics, 37–38: 3–8, doi:10.1016/0166-218X(92)90120-Y Bermond, Jean-Claude; Delorme, Charles; Farhi, Guy (1982), "Large Graphs with Given Degree and Diameter III" (PDF), Graph Theory, Proceedings of the Conference on Graph Theory, North-Holland Mathematics Studies, vol. 62, pp. 23–31, doi:10.1016/S0304-0208(08)73544-8, ISBN 9780444864499, S2CID 118362048 Also published in Annals of Mathematics (1982) 13 23–31. Buset, Dominique (2000), "Maximal cubic graphs with diameter 4", Discrete Applied Mathematics, 101 (1–3): 53–61, doi:10.1016/S0166-218X(99)00204-8 Canale, Eduardo; Rodríguez, Alexis (2012), On the application of voltage graphs to the degree/diameter problem (PDF), archived from the original (PDF) on 2020-09-28 Comellas, Francesc; Gómez, José (1994). "New Large Graphs with Given Degree and Diameter". arXiv:math/9411218. Comellas, Francesc (2024). "Table of large graphs with given degree and diameter". arXiv:2406.18994 [math.CO]. Conder, Marston (2006). "Trivalent (cubic) symmetric graphs on up to 2048 vertices". Delorme, Charles; Farhi, Guy (1984), "Large Graphs with Given Degree and Diameter - Part I", IEEE Transactions on Computers, 33 (9): 857–860, Bibcode:1984ITCmp.100..857D, doi:10.1109/TC.1984.1676504 Delorme, Charles (1985a), "Grands Graphes de Degré et Diamètre Donnés", European Journal of Combinatorics, 6 (4): 291–302, doi:10.1016/S0195-6698(85)80043-3 Delorme, Charles (1985b), "Large bipartite graphs with given degree and diameter", Journal of Graph Theory, 9 (3): 325–334, doi:10.1002/jgt.3190090304, S2CID 21199003 Dinneen, Michael J.; Hafner, Paul R. (1994), "New Results for the Degree/Diameter Problem", Networks, 24 (7): 359–367, arXiv:math/9504214, doi:10.1002/net.3230240702, S2CID 26375247 Doty, Karl (1982), "Large regular interconnection networks", Proceedings of the 3rd International Conference on Distributed Computing Systems, IEEE Computer Society, pp. 312–317 Elspas, Bernard (1964), "Topological constraints on interconnection-limited logic", 1964 Proceedings of the Fifth Annual Symposium on Switching Circuit Theory and Logical Design, pp. 133–137, doi:10.1109/SWCT.1964.27 Gómez, José (2009), "Some new large (Δ, 3)-graphs", Networks, 53 (1): 1–5, doi:10.1002/NET.V53:1 Gómez, José; Fiol, Miquel (1985), "Dense compound graphs", Ars Combinatoria, 20: 211–237 Gómez, José; Fiol, Miquel; Serra, Oriol (1993), "On large (Δ,D)-graphs", Discrete Mathematics, 114 (1–3): 219–235, doi:10.1016/0012-365X(93)90368-4 Hoffman, Alan J.; Singleton, Robert R. (1960), "Moore graphs with diameter 2 and 3", IBM Journal of Research and Development, 5 (4): 497–504, doi:10.1147/rd.45.0497, MR 0140437 Loz, Eyal; Širáň, Jozef (2008), "New record graphs in the degree-diameter problem" (PDF), Australasian Journal of Combinatorics, 41: 63–80 McKay, Brendan D.; Miller, Mirka; Širáň, Jozef (1998), "A note on large graphs of diameter two and given maximum degree", Journal of Combinatorial Theory, Series B, 74 (4): 110–118, doi:10.1006/jctb.1998.1828 Miller, Mirka; Širáň, Jozef (2013), "Moore graphs and beyond: A survey of the degree/diameter problem", Electronic Journal of Combinatorics, Dynamic survey D Molodtsov, Sergey (2006), General Theory of Information Transfer and Combinatorics, Springer, pp. 853–857, ISBN 978-3-540-46244-6 Pineda-Villavicencio, Guillermo; Gómez, José; Miller, Mirka; Pérez-Rosés, Hebert (2006), "New Largest Graphs of Diameter 6", Electronic Notes in Discrete Mathematics, 24: 153–160, doi:10.1016/j.endm.2006.06.044, hdl:1959.17/67691 Sampels, Michael (1997), "Large Networks with Small Diameter", Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, vol. 1335, Springer, Berlin, Heidelberg, pp. 288–302, doi:10.1007/BFb0024505, ISBN 978-3-540-69643-8 Storwick, Robert (1970), "Improved Construction Techniques for (d, k) Graphs", IEEE Transactions on Computers, C-19 (12): 1214–1216, Bibcode:1970ITCmp.100.1214S, doi:10.1109/T-C.1970.222861 Wegner, Gerd (1977), Graphs with given diameter and a coloring problem (PDF), Technische Universität Dortmund, doi:10.17877/DE290R-16496

External links The Degree-Diameter Problem on CombinatoricsWiki.org. Eyal Loz's degree-diameter problem page (archived 2016.) Geoffrey Exoo's degree-diameter record graphs page (archived 2015.)

Worked examples

Example 1 — a first encounter with Table of the largest known graphs of a given diameter and maximal degree

Start with the simplest possible case. Write down what Table of the largest known graphs of a given diameter and maximal degree claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Table of the largest known graphs of a given diameter and maximal degree before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Table of the largest known graphs of a given diameter and maximal degree ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Table of the largest known graphs of a given diameter and maximal degree

In research: Table of the largest known graphs of a given diameter and maximal degree appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Table of the largest known graphs of a given diameter and maximal degree in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Table of the largest known graphs of a given diameter and maximal degree is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graphs, Lists by size, so understanding it makes those chapters shorter.
In everyday life: Look for Table of the largest known graphs of a given diameter and maximal degree outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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Frequently asked questions

What is Table of the largest known graphs of a given diameter and maximal degree in simple terms?

Why does Table of the largest known graphs of a given diameter and maximal degree matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Table of the largest known graphs of a given diameter and maximal degree?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

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