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We exhibit PD-sets for the binary and non-binary codes generated by incidence matrices of triangular graphs Tn where n≥5. [ABSTRACT FROM AUTHOR]

For k ≥ 1, the odd graph denoted by O( k), is the graph with the vertex-set Ω, the set of all k-subsets of Ω = {1, 2, ..., 2 k +1}, and any two of its vertices u and v constitute an edge [ u, v] if and only if u ∩ v = /0. In this paper the binary code generated by the adjacency matrix of O( k) is studied. The automorphism group of the code is determined, and by identifying a suitable information set, a 2-PD-set of the order of k is determined. Lastly, the relationship between the dual code from O( k) and the code from its graph-theoretical complement $\overline {O(k)} $, is investigated. [ABSTRACT FROM AUTHOR]

For any prime p, we consider p-ary linear codes obtained from the span over $\mathbb{F}_p $ of rows of incidence matrices of triangular graphs, differences of the rows and adjacency matrices of line graphs of triangular graphs. We determine parameters of the codes, minimum words and automorphism groups. We also show that the codes can be used for full permutation decoding. [ABSTRACT FROM AUTHOR]

Via representation of vertex-transitive graphs on groupoids, we show that left loops with units are factors of groups, i.e., left loops are transversals of left cosets on which it is possible to define a binary operation which allows left cancellation. [ABSTRACT FROM AUTHOR]

The successful determination of reliable protein interaction networks (PINs) in several species in the post-genomic era has hitherto facilitated the quest to understanding systems and structural properties of such networks. It is envisaged that a clearer understanding of their intrinsic topological properties would elucidate evolutionary and biological topography of organisms. This, in turn, may inform the understanding of diseases' aetiology. By analysing sub-networks that are induced in various layers identified by zones defined as distance from central proteins, we show that zones of human PINs display self-similarity patterns. What is observed at a global level is repeated at lower levels of inducement. Furthermore, it is observed that these levels of strength point to refinement and specialisations in these layers. This may point to the fact that various levels of representations in the self-similarity phenomenon offer a way of measuring and distinguishing the importance of proteins in the network. To consolidate our findings, we have also considered a gene co-expression network and a class of gene regulatory networks in the same framework. In all cases, the phenomenon is significantly evident. In particular, the truly unbiased regulatory networks show finer level of articulation of self-similarity. [ABSTRACT FROM AUTHOR]