Abstract
We study the class of G-symmetric graphs Γ with diameter 2, where G is an affine-type quasiprimitive group on the vertex set of Γ. These graphs arise from normal quotient analysis as basic graphs in the class of symmetric diameter 2 graphs. It is known that \({G \cong V \rtimes G_0}\) , where V is a finite-dimensional vector space over a finite field and G 0 is an irreducible subgroup of GL (V), and Γ is a Cayley graph on V. In particular, we consider the case where \({V = \mathbb {F}_p^d}\) for some prime p and G 0 is maximal in GL (d, p), with G 0 belonging to the Aschbacher classes \({\mathcal {C}_2, \mathcal {C}_4, \mathcal {C}_6, \mathcal {C}_7}\) , and \({\mathcal {C}_8}\) . For \({G_0 \in \mathcal {C}_i, i = 2,4,8}\) , we determine all diameter 2 graphs which arise. For \({G_0 \in \mathcal {C}_6, \mathcal {C}_7}\) we obtain necessary conditions for diameter 2, which restrict the number of unresolved cases to be investigated, and in some special cases determine all diameter 2 graphs.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Finite Geometries”.
This paper forms part of the first author’s Ph.D., which is supported by an Endeavour International Postgraduate Research Scholarship (with UPAIS) and a Samaha Top-Up Scholarship from The University of Western Australia, and also forms part of an Australian Research Council Discovery Project of the last two authors. The second author holds an Australian Research Fellowship, and the third author holds a Federation Fellowship.
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Amarra, C., Giudici, M. & Praeger, C.E. Symmetric diameter two graphs with affine-type vertex-quasiprimitive automorphism group. Des. Codes Cryptogr. 68, 127–139 (2013). https://doi.org/10.1007/s10623-012-9644-z
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DOI: https://doi.org/10.1007/s10623-012-9644-z