On the asymptotic behavior of the average geodesic distance L and the compactness CB of simple connected undirected graphs whose order approaches infinity.

The average geodesic distance L Newman (2003) and the compactness CB Botafogo (1992) are important graph indices in applications of complex network theory to real-world problems. Here, for simple connected undirected graphs G of order n, we study the behavior of L(G) and CB(G), subject to the condition that their order |V(G)| approaches infinity. We prove that the limit of L(G)/n and CB(G) lies within the interval [0;1/3] and [2/3;1], respectively.

On the limit value of compactness of some graph classes.

In this paper, we study the limit of compactness which is a graph index originally introduced for measuring structural characteristics of hypermedia. Applying compactness to large scale small-world graphs (Mehler, 2008) observed its limit behaviour to be equal 1. The striking question concerning this finding was whether this limit behaviour resulted from the specifics of small-world graphs or was simply an artefact.

Phylogenetic analyses of amino acid variation in the serpin proteins.

Phylogenetic analyses of 110 serpin protein sequences revealed clades consistent with independent phylogenetic analyses based on exon-intron structure and diagnostic amino acid sites. Trees were estimated by maximum likelihood, neighbor joining, and partial split decomposition using both the BLOSUM 62 and Jones-Taylor-Thornton substitution matrices. Neighbor-joining trees gave results closest to those based on independent analyses using genomic and chromosomal data.

Vertebrate serpins: construction of a conflict-free phylogeny by combining exon-intron and diagnostic site analyses.

A combination of three independent biological features, genomic organization, diagnostic amino acid sites, and rare indels, was used to elucidate the phylogeny of the vertebrate serpin (serine protease inhibitor) superfamily. A strong correlation between serpin gene families displaying (1) a conserved exon-intron pattern and (2) family-specific combinations of amino acid residues at specific sites suggests that present-day vertebrates encompass six serpin gene families which evolved from primordial genes by massive intron insertion before or during early vertebrate radiation. Introns placed at homologous positions in the gene sequences in combination with diagnostic sequence characters may also constitute a reliable kinship indicator for other protein superfamilies.