A vertex subset F ⊆ V(G) is called a cyclic vertex-cut set of a connected graph G if G-F is disconnected such that at least two components in G-F contain cycles. The cyclic vertex connectivity is the cardinality of a minimum cyclic vertex-cut set. In this paper, we show that the cyclic vertex connectivity of the trivalent Cayley graphs TGn is equal to eight for n ≥ 4.

In this paper, we study the feedback vertex set problem for trivalent Cayley graphs, and construct a minimum feedback vertex set in trivalent Cayley graphs using the result on cube-connected cycles and the Cayley graph representation of trivalent Cayley graphs.

We propose a new class of binary nonlinear codes of constant weights derived from a permutation representation of a group that is given by a combinatorial definition such as Cayley graphs of a group. These codes are constructed by the following direct interpretation method from a group: (1) take one discrete group whose elements are defined by generators and their relations, such as those in the form of Cayley graphs; and (2) embedding the group into a binary space using some of their permutation representations by providing the generators with realization of permutations of some terms. The proposed codes are endowed with some good characteristics as follows: (a) we can easily learn information about the distances of the obtained codes, and moreover, (b) we can establish a decoding method for them that can correct random errors whose distances from code words are less than half of the minimum distances achieved using only parity checking procedures.

A minimum feedback node set in a graph is a minimum node subset whose deletion makes the graph acyclic. Its detection in a dependency graph is important to recover from a deadlock configuration. A livelock configuration is also avoidable if a check point is set in each node in the minimum feedback node set. Hence, its detection is very important to establish dependable network systems. In this letter, we give a minimum feedback node set in a trivalent Cayley graph. Assuming that each word has n bits, for any node, we can judge if it is included in the set or not in constant time.

In this paper, we investigate self diagnosable systems on multi-processor systems, known as one-step t-diagnosable systems introduced by Preparata et al. Kohda has proposed "highly structured system" to design diagnosable systems such that faulty processors are diagnosed efficiently. On the other hand, it is known that Cayley graphs have been investigated as good models for architectures of large-scale parallel processor systems. We investigate some conditions for Cayley graphs to be topologies for optimal highly structured diagnosable systems, and present several examples of optimal diagnosable systems represented by Cayley graphs.

Dynamical theory of cellular automata on groups is developed. Main results are non-Euclidean extensions of Sato and Honda's results on the dynamics of Euclidean cellular automata. The notion of the period of a configuration is redefined in a more group theoretical way. The notion of a co-finite configuration substitutes the notion of a periodic configuration, where the new term is given to it to reflect and emphasize the importance of finiteness involved. With these extended or substituted notions, the relations among period preservablity, injectivity, and Poisson stability of parallel maps are established. Residually finite groups are shown to give a nice topological property that co-finite configurations are dense in the configuration space.