Abelian Cayley Digraphs with Asymptotically Large Order for Any Given Degree

Abstract

Abelian Cayley digraphs can be constructed by using a generalization to Z n of the concept of congruence in Z. Here we use this approach to present a family of such digraphs, which, for every fixed value of the degree, have asymptotically large number of vertices as the diameter increases. Up to now, the best known asymptotically dense results were all non-constructive.

An improved Moore bound and some new optimal families of mixed Abelian Cayley graphs

“…This problem has been considered for different families of graphs. For instance: bipartite graphs in Dalfó, Fiol, and López [3]; planar graphs in Fellows, Hell, and Seyfarth [7] and in Tischenko [20]; vertex-transitive graphs in Machbeth, Šiagiová, Širáň, and Vetrík [13], and in Šiagiová and Vetrík [18]; Cayley graphs also in [13,18], and in Vetrík [21]; Cayley graphs of Abelian groups in Dougherty and Faber [6], and Aguiló, Fiol and Pérez [1]; and circulant graphs in Wong and Coppersmith [22], Morillo, Fiol, and Fàbrega [16], Fiol, Yebra, Alegre, and Valero [10], and in Monakhova [15]. For more information, see the comprehensive survey of Miller and Širáň [14].…”

An improved Moore bound and some new optimal families of mixed Abelian Cayley graphs

“…This problem has been considered for different families of graphs. For instance: bipartite graphs in Dalfó, Fiol, and López [3]; planar graphs in Fellows, Hell, and Seyfarth [7], and in Tischenko [21]; vertex-transitive graphs in Machbeth, Šiagiová, Širáň, and Vetrík [13], and in Šiagiová and Vetrík [19]; Cayley graphs also in [13,19], and in Vetrík [22]; Cayley graphs of Abelian groups in Dougherty and Faber [6], and Aguiló, Fiol and Pérez [1]; and circulant graphs in Wong and Coppersmith [23], Morillo, Fiol, and Fàbrega [17], Fiol, Yebra, Alegre, and Valero [10], and in Monakhova [16]. For more information, see the comprehensive survey of Miller and Širáň [14].…”

New results on the degree/diameter problem of mixed Abelian Cayley graphs

“…Moreover, we remark that in some cases there are other generators and/or groups that produce non-isomorphic mixed graphs with the same degree, diameter and order, than the ones given in Table 2. For instance, Z 2 × Z 16 with any of the three following sets {(1, 11), (0, 1), (0, 8)}, { (1,11), (1,0), (1,8)} or {(1, 11), (0, 5), (1, 8)} as generators, produces an optimal mixed graph for k = 6. Another example is given in Figure 6, where we show an alternative mixed graph with diameter 3x = 6, maximum order 32 and generators 1, 10, 16 (different from the one corresponding to 5, 2, 16, provided by Table 2).…”