Mixed Moore Cayley Graphs

Abstract

The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. There has been much recent interest in the problem for mixed graphs, where we allow both undirected edges and directed arcs in the graph. For a diameter 2 graph with maximum undirected degree r and directed out-degree z, a straightforward counting argument yields an upper bound M (z, r, 2) = (z + r) 2 + z + 1 for the order of the graph. Apart from the case r = 1, the only three known examples of mixed graphs attaining this bound are Cayley graphs, and there are an infinite number of feasible pairs (r, z) where the existence of mixed Moore graphs with these parameters is unknown. We use a combination of elementary group-theoretical arguments and computational techniques to rule out the existence of further examples of mixed Cayley graphs attaining the Moore bound for all orders up to 485. PreliminariesThe degree-diameter problem has its roots in the efficient design of interconnection networks. We try to find the maximum possible number of vertices in a graph where we constrain both the largest degree of any vertex and the diameter of the graph. For more information on the history and development of the degree-diameter problem, see the survey [11]. The degree-diameter problem is typically studied in both the undirected and directed cases. Recently, there has been much interest in the problem as it is applied to mixed graphs, where we allow both undirected edges and directed arcs in the graph.In the undirected case, an upper bound for the largest possible order of a graph of maximum degree d > 2 and diameter k > 1 is the Moore bound [11]:Mathematics subject classification: 05C25, 05C35

Moore mixed graphs from Cayley graphs

“…Apart from the case r = 1, only three Moore digraphs are known, which are also Cayley graphs. 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].…”

“…The first part of this paper is a brief survey about known Moore (and almost Moore) general mixed graphs that turn out to be Cayley graphs. More information about Cayley Moore mixed graphs can be found in the paper by Erskine [9]. Our main contribution is described in the second part of the paper, where we give new results on the bipartite case.…”

“…Similarly, in the case of regular graphs or digraphs, the (r, z, k) problem for mixed graphs consists of finding the largest possible number of vertices N (r, z, k) in a mixed graph with maximum undirected degree r, maximum directed out-degree z, and diameter k. For more results on this problem on graphs (and mixed graphs), see the comprehensive survey by Miller and Širáň [18]. For more results on mixed graphs, see Buset, López, and Miret [4], Dalfó [5], Dalfó, Fiol, and López [6], Erskine [9], Jørgensen [14], López, Pérez-Rosés, and Pujolàs [17], Nguyen, Miller, and Gimbert [19], and Tuite and Erskine [20].…”

“…Theorem 3.2 (Erskine [9]). The only Moore Cayley (r, z, 2)-mixed graphs with order N ≤ 485 are the following:…”

On networks with order close to the Moore bound

“…By carrying out a computer search for Cayley mixed graphs that meet the Moore bound Jørgensen found two Cayley mixed Moore graphs with undirected degree r = 3, directed out-degree z = 7, diameter k = 2 and order n = 108 [24]. However, it has been shown that there are no further Cayley mixed Moore graphs with diameter two and order ≤ 485 [17,28]. A search using a SAT solver has also completely ruled out the existence of mixed Moore graphs with diameter two and orders 40, 54 or 84 [26].…”