A proper edge colouring f of a graph G is called acyclic if there are no bichromatic cycles in the graph. The acyclic edge chromatic number or acyclic chromatic index, denoted by , is the minimum number of colours in ...A proper edge colouring f of a graph G is called acyclic if there are no bichromatic cycles in the graph. The acyclic edge chromatic number or acyclic chromatic index, denoted by , is the minimum number of colours in an acyclic edge colouring of G. In this paper, we discuss the acyclic edge colouring of middle, central, total and line graphs of prime related star graph families. Also exact values of acyclic chromatic indices of such graphs are derived and some of their structural properties are discussed.展开更多
Network is considered naturally as a wide range of different contexts, such as biological systems, social relationships as well as various technological scenarios. Investigation of the dynamic phenomena taking place i...Network is considered naturally as a wide range of different contexts, such as biological systems, social relationships as well as various technological scenarios. Investigation of the dynamic phenomena taking place in the network, determination of the structure of the network and community and description of the interactions between various elements of the network are the key issues in network analysis. One of the huge network structure challenges is the identification of the node(s) with an outstanding structural position within the network. The popular method for doing this is to calculate a measure of centrality. We examine node centrality measures such as degree, closeness, eigenvector, Katz and subgraph centrality for undirected networks. We show how the Katz centrality can be turned into degree and eigenvector centrality by considering limiting cases. Some existing centrality measures are linked to matrix functions. We extend this idea and examine the centrality measures based on general matrix functions and in particular, the logarithmic, cosine, sine, and hyperbolic functions. We also explore the concept of generalised Katz centrality. Various experiments are conducted for different networks generated by using random graph models. The results show that the logarithmic function in particular has potential as a centrality measure. Similar results were obtained for real-world networks.展开更多
We report a bioinformatic analysis of the datasets of sequences of all ten genes from the 2009 H1N1 influenza A pandemic in the state of Wisconsin. The gene with the greatest summed information entropy was found to be...We report a bioinformatic analysis of the datasets of sequences of all ten genes from the 2009 H1N1 influenza A pandemic in the state of Wisconsin. The gene with the greatest summed information entropy was found to be the hemagglutinin (HA) gene. Based upon the viral ID identifier of the HA gene sequence, the sequences of all of the genes were sorted into two subsets, depending upon whether the nucleotide occupying the position of maximum entropy, position 658 of the HA sequence, was either A or U. It was found that the information entropy (H) distributions of subsets differed significantly from each other, from H distributions of randomly generated subsets and from the H distributions of the complete datasets of each gene. Mutual information (MI) values facilitated identification of nine nucleotide positions, distributed over seven of the influenza genes, at which the nucleotide subsets were disjoint, or almost disjoint. Nucleotide frequencies at these nine positions were used to compute mutual information values that subsequently served as weighting factors for edges in a graph net-work. Seven of the nucleotide positions in the graph network are sites of synonymous mutations. Three of these sites of synonymous mutation are within a single gene, the M1 gene, which occupied the position of greatest graph centrality. It is proposed that these bioinformatic and network graph results may reflect alterations in M1-mediated viral packaging and exteriorization, known to be susceptible to synonymous mutations.展开更多
<div style="text-align:justify;"> We know that functional and structural organization is altered in human brain network due to Alzheimer’s disease. In this paper we highlight how Graph Theory techniqu...<div style="text-align:justify;"> We know that functional and structural organization is altered in human brain network due to Alzheimer’s disease. In this paper we highlight how Graph Theory techniques, its structural parameters like connectivity, diameter, vertex centrality, betweenness centrality, clustering coefficient, degree distribution, cluster analysis and graph cores are involved to analyse magnetoencephalography data to explore functional network integrity in Alzheimer’s disease affected patients. We also record that both weighted and unweighted undirected/directed graphs depending on functional connectivity analysis with attention to connectivity of the network and vertex centrality, could model and provide explanation to loss of links, status of the hub in the region of parietal, derailed synchronization in network and centrality loss at the vital left temporal region that is clinically significant were found in cases carrying Alzheimer’s disease. We also notice that graph theory driven measures such as characteristic path length and clustering coefficient could be used to study and report a sudden electroencephalography effect in Alzheimer’s disease through entropy of the cross-sample. We also provide adequate literature survey to demonstrate the latest and advanced graphical tools for both graph layouts and graph visualization to understand the complex brain networks and to unravel the mysteries of Alzheimer’s disease. </div>展开更多
文摘A proper edge colouring f of a graph G is called acyclic if there are no bichromatic cycles in the graph. The acyclic edge chromatic number or acyclic chromatic index, denoted by , is the minimum number of colours in an acyclic edge colouring of G. In this paper, we discuss the acyclic edge colouring of middle, central, total and line graphs of prime related star graph families. Also exact values of acyclic chromatic indices of such graphs are derived and some of their structural properties are discussed.
文摘Network is considered naturally as a wide range of different contexts, such as biological systems, social relationships as well as various technological scenarios. Investigation of the dynamic phenomena taking place in the network, determination of the structure of the network and community and description of the interactions between various elements of the network are the key issues in network analysis. One of the huge network structure challenges is the identification of the node(s) with an outstanding structural position within the network. The popular method for doing this is to calculate a measure of centrality. We examine node centrality measures such as degree, closeness, eigenvector, Katz and subgraph centrality for undirected networks. We show how the Katz centrality can be turned into degree and eigenvector centrality by considering limiting cases. Some existing centrality measures are linked to matrix functions. We extend this idea and examine the centrality measures based on general matrix functions and in particular, the logarithmic, cosine, sine, and hyperbolic functions. We also explore the concept of generalised Katz centrality. Various experiments are conducted for different networks generated by using random graph models. The results show that the logarithmic function in particular has potential as a centrality measure. Similar results were obtained for real-world networks.
文摘We report a bioinformatic analysis of the datasets of sequences of all ten genes from the 2009 H1N1 influenza A pandemic in the state of Wisconsin. The gene with the greatest summed information entropy was found to be the hemagglutinin (HA) gene. Based upon the viral ID identifier of the HA gene sequence, the sequences of all of the genes were sorted into two subsets, depending upon whether the nucleotide occupying the position of maximum entropy, position 658 of the HA sequence, was either A or U. It was found that the information entropy (H) distributions of subsets differed significantly from each other, from H distributions of randomly generated subsets and from the H distributions of the complete datasets of each gene. Mutual information (MI) values facilitated identification of nine nucleotide positions, distributed over seven of the influenza genes, at which the nucleotide subsets were disjoint, or almost disjoint. Nucleotide frequencies at these nine positions were used to compute mutual information values that subsequently served as weighting factors for edges in a graph net-work. Seven of the nucleotide positions in the graph network are sites of synonymous mutations. Three of these sites of synonymous mutation are within a single gene, the M1 gene, which occupied the position of greatest graph centrality. It is proposed that these bioinformatic and network graph results may reflect alterations in M1-mediated viral packaging and exteriorization, known to be susceptible to synonymous mutations.
文摘<div style="text-align:justify;"> We know that functional and structural organization is altered in human brain network due to Alzheimer’s disease. In this paper we highlight how Graph Theory techniques, its structural parameters like connectivity, diameter, vertex centrality, betweenness centrality, clustering coefficient, degree distribution, cluster analysis and graph cores are involved to analyse magnetoencephalography data to explore functional network integrity in Alzheimer’s disease affected patients. We also record that both weighted and unweighted undirected/directed graphs depending on functional connectivity analysis with attention to connectivity of the network and vertex centrality, could model and provide explanation to loss of links, status of the hub in the region of parietal, derailed synchronization in network and centrality loss at the vital left temporal region that is clinically significant were found in cases carrying Alzheimer’s disease. We also notice that graph theory driven measures such as characteristic path length and clustering coefficient could be used to study and report a sudden electroencephalography effect in Alzheimer’s disease through entropy of the cross-sample. We also provide adequate literature survey to demonstrate the latest and advanced graphical tools for both graph layouts and graph visualization to understand the complex brain networks and to unravel the mysteries of Alzheimer’s disease. </div>
