Maximum eigenvalues of the reciprocal distance matrix and the reverse Wiener matrix

2008 ◽  
Vol 108 (5) ◽  
pp. 858-864 ◽  
Author(s):  
Bo Zhou ◽  
Nenad Trinajstić
Keyword(s):  
Distance Matrix ◽  
Reciprocal Distance Matrix

scholarly journals On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 512
Author(s):  
Maryam Baghipur ◽  
Modjtaba Ghorbani ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Second Largest Eigenvalue ◽  
Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

scholarly journals The Wiener Index of Circulant Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Houqing Zhou
Keyword(s):  
Distributed Computing ◽  
Interconnection Networks ◽  
Distance Matrix ◽  
Wiener Index ◽  
Important Class ◽  
Circulant Graphs ◽  
Parallel And Distributed Computing ◽  
Harary Index ◽  
Largest Eigenvalues ◽  
Reciprocal Distance Matrix

Circulant graphs are an important class of interconnection networks in parallel and distributed computing. In this paper, we discuss the relation of the Wiener index and the Harary index of circulant graphs and the largest eigenvalues of distance matrix and reciprocal distance matrix of circulants. We obtain the following consequence:W/λ=H/μ;2W/n=λ;2H/n=μ, whereW,Hdenote the Wiener index and the Harary index andλ,μdenote the largest eigenvalues of distance matrix and reciprocal distance matrix of circulant graphs, respectively. Moreover we also discuss the Wiener index of nonregular graphs with cut edges.

THE HARARY INDEX OF A GRAPH UNDER PERTURBATION

2010 ◽  
Vol 02 (02) ◽  
pp. 247-255 ◽  
Author(s):  
CHANG-XIANG HE ◽  
PING CHEN ◽  
BAO-FENG WU
Keyword(s):  
Distance Matrix ◽  
Harary Index ◽  
Reciprocal Distance Matrix

The Harary index is defined as the half-sum of the elements in the reciprocal distance matrix. In this paper, we investigate how the Harary index behaves when the graph is perturbed by grafting or moving edges.

Bounds and extremal graphs for Harary energy

2021 ◽  
pp. 2150149
Author(s):  
A. Alhevaz ◽  
M. Baghipur ◽  
H. A. Ganie ◽  
K. C. Das
Keyword(s):  
Connected Graph ◽  
Distance Matrix ◽  
Upper Bounds ◽  
Extremal Graphs ◽  
Lower And Upper Bounds ◽  
Energy Formula ◽  
Matrix Formula ◽  
The Absolute ◽  
Graph Parameters ◽  
Reciprocal Distance Matrix

Let [Formula: see text] be a connected graph of order [Formula: see text] and let [Formula: see text] be the reciprocal distance matrix (also called Harary matrix) of the graph [Formula: see text]. Let [Formula: see text] be the eigenvalues of the reciprocal distance matrix [Formula: see text] of the connected graph [Formula: see text] called the reciprocal distance eigenvalues of [Formula: see text]. The Harary energy [Formula: see text] of a connected graph [Formula: see text] is defined as sum of the absolute values of the reciprocal distance eigenvalues of [Formula: see text], that is, [Formula: see text] In this paper, we establish some new lower and upper bounds for [Formula: see text] in terms of different graph parameters associated with the structure of the graph [Formula: see text]. We characterize the extremal graphs attaining these bounds. We also obtain a relation between the Harary energy and the sum of [Formula: see text] largest adjacency eigenvalues of a connected graph.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals The Diversity, Composition, and Metabolic Pathways of Archaea in Pigs

Animals ◽  
2021 ◽  
Vol 11 (7) ◽  
pp. 2139
Author(s):  
Feilong Deng ◽  
Yushan Li ◽  
Yunjuan Peng ◽  
Xiaoyuan Wei ◽  
Xiaofan Wang ◽  
...  
Keyword(s):  
Relative Abundance ◽  
Antibiotic Resistance Genes ◽  
Alpha Diversity ◽  
Distance Matrix ◽  
Archaeal Community ◽  
Gene Families ◽  
Shannon Index ◽  
Metabolic Potential ◽  
Substantial Progress ◽  
Archaeal Communities

Archaea are an essential class of gut microorganisms in humans and animals. Despite the substantial progress in gut microbiome research in the last decade, most studies have focused on bacteria, and little is known about archaea in mammals. In this study, we investigated the composition, diversity, and functional potential of gut archaeal communities in pigs by re-analyzing a published metagenomic dataset including a total of 276 fecal samples from three countries: China (n = 76), Denmark (n = 100), and France (n = 100). For alpha diversity (Shannon Index) of the archaeal communities, Chinese pigs were less diverse than Danish and French pigs (p < 0.001). Consistently, Chinese pigs also possessed different archaeal community structures from the other two groups based on the Bray–Curtis distance matrix. Methanobrevibacter was the most dominant archaeal genus in Chinese pigs (44.94%) and French pigs (15.41%), while Candidatus methanomethylophilus was the most predominant in Danish pigs (15.71%). At the species level, the relative abundance of Candidatus methanomethylophilus alvus, Natrialbaceae archaeon XQ INN 246, and Methanobrevibacter gottschalkii were greatest in Danish, French, and Chinese pigs with a relative abundance of 14.32, 11.67, and 16.28%, respectively. In terms of metabolic potential, the top three pathways in the archaeal communities included the MetaCyc pathway related to the biosynthesis of L-valine, L-isoleucine, and isobutanol. Interestingly, the pathway related to hydrogen consumption (METHANOGENESIS-PWY) was only observed in archaeal reads, while the pathways participating in hydrogen production (FERMENTATION-PWY and PWY4LZ-257) were only detected in bacterial reads. Archaeal communities also possessed CAZyme gene families, with the top five being AA3, GH43, GT2, AA6, and CE9. In terms of antibiotic resistance genes (ARGs), the class of multidrug resistance was the most abundant ARG, accounting for 87.41% of archaeal ARG hits. Our study reveals the diverse composition and metabolic functions of archaea in pigs, suggesting that archaea might play important roles in swine nutrition and metabolism.

Precise clustering analysis of Internet financial credit reporting dependent on multidimensional attribute sparse large data

2021 ◽  
pp. 002072092110020
Author(s):  
Lingling Chen ◽  
Yuanyuan Zhang ◽  
Min Zeng
Keyword(s):  
Clustering Analysis ◽  
Large Data ◽  
Distance Matrix ◽  
The Internet ◽  
Relationship Matrix ◽  
Clustering Methods ◽  
Correlation Relationship ◽  
Credit Reporting ◽  
Financial Credit ◽  
Approximate Distance

Given that the traditional methods cannot perform clustering analysis on the Internet financial credit reporting directly and effectively, a kind of precise clustering analysis of internet financial credit reporting dependent on multidimensional attribute sparse large data is proposed. By measuring the overall distance between Internet financial credit reporting through the sparse large data with multidimensional attributes, the multidimensional attribute sparse large data are used to perform clustering analysis on the overall distance matrix and the component approximate distance matrix between the data, respectively. The correlation relationship between the Internet financial credit reporting under these two perspectives is taken into comprehensive consideration. Multidimensional attribute sparse large data pairs are used to reflect the comprehensive relationship matrix of the original Internet financial credit reporting to achieve clustering with relatively high quality. Numerical experiments show that compared with the traditional clustering methods, the method proposed in this paper can not only reflect the overall data features effectively, but also improve the clustering effect of the original Internet financial credit reporting data through the analysis of the correlation relationship between the important component attribute sequences.

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