A Note on Moore Cayley Digraphs

Gavrilyuk, Alexander L.; Hirasaka, Mitsugu; Kabanov, V. V.

doi:10.1007/s00373-021-02286-w

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2023

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“…Namely, the already mentioned Bosák digraph, and the two digraphs of Jørgensen [14]; see later Theorem 3.2 by Erskine [9]. Some proofs of the non-existence of Cayley Moore digraphs for some order values have been given by López, Pérez-Rosés, and Pujolàs [17], López, Miret, and Fernández [16] , Erskine [9], and Gavrilyuk, Hirasaka, and Kabanov [12].…”

Section: Preliminariesmentioning

confidence: 98%

Moore mixed graphs from Cayley graphs

Dalfó¹,

Fiol²

2023

EJGTA

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A Moore (r, z, k)-mixed graph G has every vertex with undirected degree r, directed in-and outdegree z, diameter k, and number of vertices (or order) attaining the corresponding Moore bound M (r, z, k) for mixed graphs. When the order of G is close to M (r, z, k) vertices, we refer to it as an almost Moore graph. The first part of this paper is a survey about known Moore (and almost Moore) general mixed graphs that turn out to be Cayley graphs. Then, in the second part of the paper, we give new results on the bipartite case. First, we show that Moore bipartite mixed graphs with diameter three are distance-regular, and their spectra are fully characterized. In particular, an infinity family of Moore bipartite (1, z, 3)-mixed graphs is presented, which are Cayley graphs of semidirect products of groups. Our study is based on the line digraph technique, and on some results about when the line digraph of a Cayley digraph is again a Cayley digraph.

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