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A Note on Moore Cayley Digraphs

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Abstract

Let \(\varDelta\) be a digraph of diameter 2 with the maximum undirected vertex degree t and the maximum directed out-degree z. The largest possible number v of vertices of \(\varDelta\) is given by the following generalization of the Moore bound:

$$\begin{aligned} v\le (z+t)^2+z+1, \end{aligned}$$

and a digraph attaining this bound is called a Moore digraph. Apart from the case \(t=1\), only three Moore digraphs are known, which are also Cayley graphs. Using computer search, Erskine (J Interconnect Netw, 17: 1741010, 2017) ruled out the existence of further examples of Cayley digraphs attaining the Moore bound for all orders up to 485. We use an algebraic approach to this problem, which goes back to an idea of G. Higman from the theory of association schemes, also known as Benson’s Lemma in finite geometry, and show non-existence of Moore Cayley digraphs of certain orders.

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References

  1. Aschbacher, M.: The nonexistence of rank three permutation groups of degree \(3250\) and subdegree \(57\). J. Algebra 19(3), 538–540 (1971)

    Article  MathSciNet  Google Scholar 

  2. Bannai, E., Ito, T.: On finite Moore graphs. J. Fac. Sci. Univ. Tokyo Math. 20, 191–208 (1973)

    MathSciNet  MATH  Google Scholar 

  3. Belousov, I.N., Makhnev, A.A.: On automorphisms of the generalized hexagon of order \((3,27)\). Proc. Steklov Inst. Math. 267(1), 33–43 (2009)

    Article  MathSciNet  Google Scholar 

  4. Benson, C.T.: On the structure of generalized quadrangles. J. Algebra 15, 443–454 (1970)

    Article  MathSciNet  Google Scholar 

  5. Bosák, J.: Partially directed Moore graphs. Math. Slov. 29(2), 181–196 (1979)

    MathSciNet  MATH  Google Scholar 

  6. Bridges, W.G., Toueg, S.: On the impossibility of directed Moore graphs. J. Comb. Theory Ser. B 29(3), 339–341 (1980)

    Article  MathSciNet  Google Scholar 

  7. Brunat, J.M., Espona, M., Fiol, M.A., Serra, O.: On Cayley line digraphs. Discrete Math. 138(1), 147–159 (1995)

    Article  MathSciNet  Google Scholar 

  8. Cameron, P.J.: Permutation Groups, volume 45 of London Math. Soc., Student Texts. Cambridge Univ. Press, Cambridge (1999)

    Google Scholar 

  9. Damerell, R.M.: On Moore graphs. Math. Proc. Camb. Philos. Soc. 74, 227–236 (1973)

    Article  MathSciNet  Google Scholar 

  10. De Winter, S., Kamischke, E., Wang, Z.: Automorphisms of strongly regular graphs with applications to partial difference sets. Designs Code Cryptogr. 79, 471–485 (2016)

    Article  MathSciNet  Google Scholar 

  11. Duval, A.M.: A directed graph version of strongly regular graphs. J. Comb. Theory Ser. A 47, 71–100 (1988)

    Article  MathSciNet  Google Scholar 

  12. Erskine, G.: Mixed Moore Cayley graphs. J. Interconnect. Netw. 17(3–4), 1741010 (2017)

    Article  Google Scholar 

  13. Feit, W., Higman, G.: The nonexistence of certain generalized polygons. J. Algebra 1(2), 114–131 (1964)

    Article  MathSciNet  Google Scholar 

  14. Gavrilyuk, A.L., Makhnev, A.A.: On automorphisms of distance-regular graphs with intersection array \(\{56,45,1;1,9,56\}\). Doklady Math. 81(3), 439–442 (2010)

    Article  MathSciNet  Google Scholar 

  15. Gimbert, J.: Enumeration of almost Moore digraphs of diameter two. Discrete Math. 231(1–3), 177–190 (2001)

    Article  MathSciNet  Google Scholar 

  16. Godsil, C.D., Hobart, S.A., Martin, W.J.: Representations of directed strongly regular graphs. Eur. J. Combin. 28, 1980–1993 (2007)

    Article  MathSciNet  Google Scholar 

  17. Herzog, M.: On finite simple groups of order divisible by three primes only. J. Algebra 10, 383–388 (1968)

    Article  MathSciNet  Google Scholar 

  18. Hoffman, A.J., Singleton, R.R.: On Moore graphs with diameters 2 and 3. IBM J. Res. Dev. 4(5), 497–504 (1960)

    Article  MathSciNet  Google Scholar 

  19. Isaacs, I.M.: Algebra: a Graduate Course. Graduate Studies in Mathematics, vol. 100. American Mathematical Society, (1994)

    Google Scholar 

  20. Jørgensen, L.K.: Non-existence of directed strongly regular graphs. Discrete Math. 264, 111–126 (2003)

    Article  MathSciNet  Google Scholar 

  21. Jørgensen, L.K.: New mixed Moore graphs and directed strongly regular graphs. Discrete Math. 338(6), 1011–1016 (2015)

    Article  MathSciNet  Google Scholar 

  22. L. K. Jørgensen. Automorphisms of a strongly regular graph \((162,21,0,3)\), 2018. In: Talk at Symmetry vs Regularity: the first 50 years since Weisfeiler-Leman stabilization, July 1–July 7, 2018, Pilsen, Czech Republic

  23. López, N., Miret, J.M., Fernández, C.: Non existence of some mixed Moore graphs of diameter \(2\) using SAT. Discrete Math. 339(2), 589–596 (2016)

    Article  MathSciNet  Google Scholar 

  24. Makhnev, A.A., Paduchikh, D.V.: On the automorphism group of the Aschbacher graph. Proc. Steklov Inst. Math. 267, 149–63 (2009)

    Article  MathSciNet  Google Scholar 

  25. Makhnev, A.A., Paduchikh, D.V., Tsiovkina, L.Y.: Edge-symmetric distance-regular coverings of cliques: the affine case. Sib. Math. J. 54(6), 1076–1087 (2013)

    Article  MathSciNet  Google Scholar 

  26. Mačaj, M., Širáň, J.: Search for properties of the missing Moore graph. Linear Algebra Appl. 432, 2381–2398 (2010)

    Article  MathSciNet  Google Scholar 

  27. Nguyen, M.H., Miller, M., Gimbert, J.: On mixed Moore graphs. Discrete Math. 307(7), 964–970 (2007)

    Article  MathSciNet  Google Scholar 

  28. Plesnik, J., Znám, S.: Strongly geodetic directed graphs. Acta FRN Univ. Comen.-Math. 29, 29–34 (1974)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous referees for their useful comments.

Alexander Gavrilyuk is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant number NRF-2018R1D1A1B07047427).

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Authors and Affiliations

  1. Pusan National University, 2, Busandaehak-ro 63beon-gil, Geumjeong-gu, Busan, 46241, Republic of Korea

    Alexander L. Gavrilyuk & Mitsugu Hirasaka

  2. Krasovskii Institute of Mathematics and Mechanics, Kovalevskaya str., 16, Yekaterinburg, 620990, Russia

    Alexander L. Gavrilyuk & Vladislav Kabanov

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  1. Alexander L. Gavrilyuk
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  2. Mitsugu Hirasaka
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  3. Vladislav Kabanov
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Correspondence to Alexander L. Gavrilyuk.

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Gavrilyuk, A.L., Hirasaka, M. & Kabanov, V. A Note on Moore Cayley Digraphs. Graphs and Combinatorics 37, 1509–1520 (2021). https://doi.org/10.1007/s00373-021-02286-w

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