Eccentricity energy change of complete multipartite graphs due to edge deletion

The eccentricity matrix ɛ(G) of a graph G is obtained from the distance matrix of G by retaining the largest distances in each row and each column, and leaving zeros in the remaining ones. The eccentricity energy of G is sum of the absolute values of the eigenvalues of ɛ(G). Although the eccentricity matrices of graphs are closely related to the distance matrices of graphs, a number of properties of eccentricity matrices are substantially different from those of the distance matrices. The change in eccentricity energy of a graph due to an edge deletion is one such property. In this article, we give examples of graphs for which the eccentricity energy increase (resp., decrease) but the distance energy decrease (resp., increase) due to an edge deletion. Also, we prove that the eccentricity energy of the complete k-partite graph Kn 1, ... , nk with k ≥ 2 and ni ≥ 2, increases due to an edge deletion.

Relationship between adjacency and distance matrix of graph of diameter two

The relationship among every pair of vertices in a graph can be represented as a matrix, such as in adjacency matrix and distance matrix. Both adjacency and distance matrices have the same property. Adjacency and distance matrices are both symmetric matrix with diagonals entries equals to 0. In this paper, we discuss relationships between adjacency matrix and distance matrix of a graph of diameter two, which is <em>D=2(J-I)-A</em>. From this relationship, we also determine the value of the determinant matrix <em>A+D</em> and the upper bound of determinant of matrix <em>D</em>.

Hobotnica: exploring molecular signature quality

A Molecular Features Set (MFS), is a result of a vast diversity of bioinformatics pipelines. The lack of a “gold standard” for most experimental data modalities makes it difficult to provide valid estimation for a particular MFS's quality. Yet, this goal can partially be achieved by analyzing inner-sample Distance Matrices (DM) and their power to distinguish between phenotypes. The quality of a DM can be assessed by summarizing its power to quantify the differences of inner-phenotype and outer-phenotype distances. This estimation of the DM quality can be construed as a measure of the MFS's quality. Here we propose Hobotnica, an approach to estimate MFSs quality by their ability to stratify data, and assign them significance scores, that allow for collating various signatures and comparing their quality for contrasting groups.

Selection of Moment Vectors in Protein Sequence Comparison Under Binary Representation

Similarity/dissimilarity study of protein and genome sequences remains a challenging task and selection of techniques and descriptors to be adopted, plays an important role in computational biology. Again, genome sequence comparison is always preferred to protein sequence comparison due the presence of 20 amino acids in protein sequence compared to only 4 nucleotides in genome sequence. So it is important to consider suitable representation that is both time and space efficient and also equally applicable to protein sequences of equal and unequal lengths. In the binary form of representation, Fourier transform of a protein sequence reduces to the transformation of 20 simple binary sequences in Fourier domain, where in each such sequence, Perseval’s Identity gives a very simple computable form of power spectrum. This gives rise to readily acceptable forms of moments of different degrees. Again such moments, when properly normalized, show a monotonically descending trend with the increase in the degrees of the moments. So it is better to stick to moments of smaller degrees only. In this paper, descriptors are taken as 20 component vectors, where each component corresponds to a general second order moment of one of the 20 simple binary sequences. Then distance matrices are obtained by using Euclidean distance as the distance measure between each pair of sequence. Phylogenetic trees are obtained from the distance matrices using UPGMA algorithm. In the present paper, the datasets used for similarity/dissimilarity study are 9 ND4, 16 ND5, 9 ND6, 24 TF proteins and 12 Baculovirus proteins. It is found that the phylogenetic trees produced by the present method are at par with those produced by the earlier methods adopted by other authors and also their known biological references. Further it takes less computational time and also it is equally applicable to sequences of equal and unequal lengths.

Split it up and see: using proxies to highlight divergent inter-populational performances in aquaculture standardised conditions

Background Considering wild inter-populational phenotypic differentiation can facilitate domestication and subsequent production of new species. However, comparing all populations across a species range to identify those exhibiting suitable key traits for aquaculture (KTA; i.e. important for domestication and subsequent production) expressions is not feasible. Therefore, proxies highlighting inter-populational divergences in KTA are needed. The use of such proxies would allow to identify, prior to bioassays, the wild population pairs which are likely to present differentiations in KTA expressions in aquaculture conditions. Here, we assessed the relevance of three alternative proxies: (i) genetic distance, (ii) habitat divergence, and (iii) geographic/hydrologic distances. We performed this evaluation on seven allopatric populations of Perca fluviatilis for which divergences in KTA had already been shown. Results We showed differences in the correlation degree between the alternative proxy-based and KTA-based distance matrices, with the genetic proxy being correlated to the highest number of KTA. However, no proxy was correlated to all inter-populational divergences in KTA. Conclusion For future domestication trials, we suggest using a multi-proxy assessment along with a prioritisation strategy to identify population pairs which are of interest for further evaluation in bioassays.

Entropy Measures of Distance Matrix

Bonchev and Trinajstic defined two distance based entropy measures to measure the molecular branching of molecular graphs in 1977 [Information theory, distance matrix, and molecular branching, J. Chem. Phys., 38 (1977), 4517&ndash;4533]. In this paper we use these entropy measures which are based on distance matrices of graphs. The first one is based on distribution of distances in distance matrix and the second one is based on distribution of distances in upper triangular submatrix. We obtain the two entropy measures of paths, stars, complete graphs, cycles and complete bipartite graphs. Finally we obtain the minimal trees with respect to these entropy measures with fixed diameter.