Algebraic and Computer-based Methods in the Undirected Degree/diameter Problem - a Brief Survey

This paper discusses the most popular algebraic techniques and computational methods that have been used to construct large graphs with given degree and diameter

LDPC codes from voltage graphs

Several well-known structure-based constructions of LDPC codes, for example codes based on permutation and circulant matrices and in particular, quasi-cyclic LDPC codes, can be interpreted via algebraic voltage assignments. We explain this connection and show how this idea from topological graph theory can be used to give simple proofs of many known properties of these codes. In addition, the notion of abelianinevitable cycle is introduced and the subgraphs giving rise to these cycles are classified. We also indicate how, by using more sophisticated voltage assignments, new classes of good LDPC codes may be obtained

On the limitations of the use of solvable groups in Cayley graph cage constructions

AbstractA (k,g)-cage is a (connected) k-regular graph of girth g having smallest possible order. While many of the best known constructions of small k-regular graphs of girth g are known to be Cayley graphs, there appears to be no general theory of the relationship between the girth of a Cayley graph and the structure of the underlying group. We attempt to fill this gap by focusing on the girth of Cayley graphs of nilpotent and solvable groups, and present a series of results supporting the intuitive notion that the closer a group is to being abelian, the less suitable it is for constructing Cayley graphs of large girth. Specifically, we establish the existence of upper bounds on the girth of Cayley graphs with respect to the nilpotency class and/or the derived length of the underlying group, when this group is nilpotent or solvable, respectively

Toxic Hazards Research Unit annual technical report, 1969 Final report, Jun. 1968 - May 1969

Apollo materials toxicity screening tests and effects of ethylene glycol, monomethylhydrazine, NF3, OF2, and ClF

Small vertex-transitive graphs of given degree and girth

We investigate the basic interplay between the small k-valent vertex-transitive graphs of girth g and the (k, g)-cages, the smallest k-valent graphs of girth g. We prove the existence of k-valent Cayley graphs of girth g for every pairof parameters k ≥ 2 and g ≥ 3, improve the lower bounds on the order of the smallest (k, g) vertex-transitive graphs forcertain families with prime power girth, and generalize the construction of Bray, Parker and Rowley that has yielded several of the smallest known (k, g)-graphs

Algebraic Design and Implementation of Protograph Codes using Non-Commuting Permutation Matrices

Random lifts of graphs, or equivalently, random permutation matrices, have been used to construct good families of codes known as protograph codes. An algebraic analog of this approach was recently presented using voltage graphs, and it was shown that many existing algebraic constructions of graph-based codes that use commuting permutation matrices may be seen as special cases of voltage graph codes. Voltage graphs are graphs that have an element of a finite group assigned to each edge, and the assignment determines a specific lift of the graph. In this paper we discuss how assignments of permutation group elements to the edges of a base graph affect the properties of the lifted graph and corresponding codes, and present a construction method of LDPC code ensembles based on noncommuting permutation matrices. We also show encoder and decoder implementations for these codes

Algebraic Design and Implementation of Protograph Codes using Non-Commuting Permutation Matrices

Random lifts of graphs, or equivalently, random permutation matrices, have been used to construct good families of codes known as protograph codes. An algebraic analog of this approach was recently presented using voltage graphs, and it was shown that many existing algebraic constructions of graph-based codes that use commuting permutation matrices may be seen as special cases of voltage graph codes. Voltage graphs are graphs that have an element of a finite group assigned to each edge, and the assignment determines a specific lift of the graph. In this paper we discuss how assignments of permutation group elements to the edges of a base graph affect the properties of the lifted graph and corresponding codes, and present a construction method of LDPC code ensembles based on noncommuting permutation matrices. We also show encoder and decoder implementations for these codes