Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
left loop; quasi-associative; Cayley graph; Hoffman--Singleton graph; vertex transitive
left loop; quasi-associative; Cayley graph; Hoffman--Singleton graph; vertex transitive
Summary:
We construct a family of vertex transitive graphs on a left loop structure of order $2q^2$ where $q$ is a power of a prime such that $q\equiv 1 \bmod 4$. The graphs are of diameter 2. The smallest of these graphs is isomorphic to the Hoffman--Singleton graph.
References:
[1] Baez K.: Towards a geometric theory for left loops. Comment. Math. Univ. Carolin. 55 (2014), no. 3, 315–323.
[2] Gauyacq G.: On quasi-Cayley graphs. Discrete Appl. Math. 77 (1997), no. 1, 43–58. DOI 10.1016/S0166-218X(97)00098-X
[3] Godsil C., Royle G.: Algebraic Graph Theory. Graduate Texts in Mathematics, 207, Springer, New York, 2001.
[4] Hafner P. R.: The Hoffman–Singleton graph and its automorphisms. J. Algebraic Combin. 18 (2003), no. 1, 7–12. DOI 10.1023/A:1025136524481
[5] Hoffman A. J., Singleton R. R.: On Moore graphs with diameters $2$ and $3$. IBM J. Res. Develop. 4 (1960), 497–504. DOI 10.1147/rd.45.0497
[6] James L. O.: A combinatorial proof that the Moore $(7,2)$ graph is unique. Utilitas Math. 5 (1974), 79–84.
[7] Mwambene E.: Characterisation of regular graphs as loop graphs. Quaest. Math. 28 (2005), no. 2, 243–250. DOI 10.2989/16073600509486125
[8] Mwambene E.: Representing vertex-transitive graphs on groupoids. Quaest. Math. 29 (2006), no. 3, 279–284. DOI 10.2989/16073600609486163 | Zbl 1107.05080
[9] Robertson G. N.: Graphs Minimal under Girth, Valency and Connectivity Constraints. PhD Thesis, University of Waterloo, Ontario, 1969.
[10] Sabidussi G.: Vertex-transitive graphs. Monatsh. Math. 68 (1964), 426–438. DOI 10.1007/BF01304186 | Zbl 0136.44608
[11] Magma Computational Algebra System. Computational Algebra Group, School of Mathematics and Statistics, University of Sydney: http://magma.maths.usyd.edu.au .
Search
Partner of
