Beyond graph energy: Norms of graphs and matrices

Nikiforov, Vladimir

doi:10.1016/j.laa.2016.05.011

Cited by 52 publications

(34 citation statements)

References 36 publications

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“…A more general definition of graph energy was suggested by Nikiforov [80][81] . Let M be a n by n real matrix and the singular values are denoted by s 1 , s 2 , … s n , the total energy, ME, obtained from M is defined by…”

Section: Characterization Of Network Motifs -Motif Graph Energy Recimentioning

confidence: 99%

Computational analysis of molecular networks using spectral graph theory, complexity measures and information theory

Huang

Tsai

Kurubanjerdjit

et al. 2019

Preprint

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Molecular networks are described in terms of directed multigraphs, so-called network motifs. Spectral graph theory, reciprocal link and complexity measures were utilized to quantify network motifs. It was found that graph energy, reciprocal link and cyclomatic complexity can optimally specify network motifs with some degree of degeneracy. Biological networks are built up from a finite number of motif patterns; hence, a graph energy cutoff exists and the Shannon entropy of the motif frequency distribution is not maximal. Also, frequently found motifs are irreducible graphs.Network similarity was quantified by gauging their motif frequency distribution functions using Jensen-Shannon entropy. This method allows us to determine the distance between two networks regardless of their nodes' identities and network sizes.This study provides a systematic approach to dissect the complex nature of biological networks. Our novel method different from any other approach. The findings support the view that there are organizational principles underlying molecular networks. Biological networks and network motifsMolecular biological networks are the basis of biological processes, where the networks exhibit biological functions through interaction among the genetic elements.Each module is expected to perform specific functions, separable from the functions of other modules 1-3 . Such modular networks can be decomposed into smaller clusters, also known as network motifs. These motifs show interesting dynamical behaviors, in which cooperativity effects between the motif components play a critical role in human diseases.Essentially, network-based analysis falls into the following major categories: (1) motif identification and analysis, (2) global architecture study, (3) local topological properties, and (4) robustness of the network under different types of perturbations.For the first category, there are a number of publicly available network motif detection tools, namely MFINDER 4 , MAVISTO 5 , FANMOD 6 , NetMatch 7 and SNAVI 8 .

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Section: Characterization Of Network Motifs -Motif Graph Energy Recimentioning

confidence: 99%

Computational analysis of molecular networks using spectral graph theory, complexity measures and information theory

Huang

Tsai

Kurubanjerdjit

et al. 2019

Preprint

View full text Add to dashboard Cite

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“…This leads to a wealth of work on energy-like graph spectral invariant in regards to, e.g., (signless) Laplacian matrix [27,28], Randić matrix [29], and distance matrix [30]. More recent results can be found in [17,18,24,[29][30][31][32][33]. In another direction, the graph energy has been extended to digraphs and various energies of digraphs such as energy [34] and skew energy [35] were put forward and extensively studied.…”

Section: Introductionmentioning

confidence: 99%

“…This concept was generalized by Nikiforov by defining the energy of any matrix [31]. Many studies, both theory-and application-oriented, have been done along this direction [24,32,34]. For example, if ξ 1 , ξ 2 , .…”

Section: Introductionmentioning

confidence: 99%

On the Generalized Distance Energy of Graphs

2019

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is the distance matrix and Tr(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.

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“…A number of extremal problems about the trace norm of matrices have been presented in the survey [4], including many upper bounds on A * . Since lower bounds on A * have not been studied in comparative detail, in this paper we initiate the study of the minimum trace norm of square (0, 1)-matrices with given number of ones.…”

Section: Introductionmentioning

confidence: 99%

“…It is not hard to see that ψ n (m) ≥ √ m; in fact, writing |A| 2 for the Frobenius norm of a matrix A, one can come up with the following simple result (see, e.g., Theorem 4.3 of [4]):…”

Section: Introductionmentioning

confidence: 99%

On the minimum trace norm/energy of (0,1)-matrices

Nikiforov

Agudelo

2017

Linear Algebra and its Applications

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The trace norm of a matrix is the sum of its singular values. This paper presents results on the minimum trace norm ψ n (m) of (0, 1)-matrices of size n × n with exactly m ones. It is shown that:(1) if n ≥ 2 and n < m ≤ 2n, then ψ n (m) ≤ m + 2 (m − 1), with equality if and only if m is a prime;(2) if n ≥ 4 and 2n < m ≤ 3n, then ψ n (m) ≤ m + 2 2 ⌊m/3⌋, with equality if and only if m is a prime or a double of a prime;(with equality if and only if there is an integer k ≥ 1 such that m = 12k ± 2 and 4k ± 1, 6k ± 1, 12k ± 1 are primes. Now, let m be a prime or a double of prime, and suppose that m = 3k + s, where 1 ≤ s ≤ 2. Applying Proposition 7 to the matrix

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Beyond graph energy: Norms of graphs and matrices

Cited by 52 publications

References 36 publications

Computational analysis of molecular networks using spectral graph theory, complexity measures and information theory

Computational analysis of molecular networks using spectral graph theory, complexity measures and information theory

On the Generalized Distance Energy of Graphs

On the minimum trace norm/energy of (0,1)-matrices

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