Enumeration of labeled, connected, homeomorphically irreducible graphs

AbstractThe convergence of the perturbation expansion for the S-matrix in interaction representation of a three-boson contact interaction is investigated. A lower bound is obtained for the integrals corresponding to irreducible graphs when the total rest mass of the system is insufficient for bare particles to be created. It is shown that in this case the perturbation expansion cannot converge no matter what value the coupling constant has. A discussion is given of the bearing of this result on the general problem of the convergence of perturbation expansions for the S-matrix in renormalized field theories.

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On the possible transition of the Gaussian model of an imperfect gas

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1977.0172 ◽

1977 ◽

Vol 357 (1690) ◽

pp. 345-353 ◽

Cited By ~ 1

Keyword(s):

Gaussian Function ◽

Gaussian Model ◽

Radius Of Convergence ◽

Virial Series ◽

Imperfect Gas ◽

Cyclomatic Number ◽

Analytic Behaviour ◽

Irreducible Graphs ◽

Limiting Case ◽

Third Derivative

It is known that the model of an imperfect gas for which the Mayer function is the Gaussian function - A exp ( - r 2 / a 2 ) of the distance r between two molecules is reducible to a problem in graph theory. It is shown that if the irreducible graphs occurring in the virial series are grouped according to their cyclomatic number we can expand the virial series in powers of the parameter A . The radius of convergence of the first few sub-series is the same. For the two-dimensional gas in the limiting case of A small, we find that, at a certain density, well outside the radius of convergence of the virial series, the third derivative of pressure vanishes. It is suggested that this may be the indication of a phase-transition of the model, and that its analytic behaviour is similar to that of lattice-type antiferromagnetic models.

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Dissecting molecular network structures using a network subgraph approach

PeerJ ◽

10.7717/peerj.9556 ◽

2020 ◽

Vol 8 ◽

pp. e9556

Author(s):

Chien-Hung Huang ◽

Efendi Zaenudin ◽

Jeffrey J.P. Tsai ◽

Nilubon Kurubanjerdjit ◽

Eskezeia Y. Dessie ◽

...

Keyword(s):

Biological Networks ◽

Spectral Graph Theory ◽

Molecular Networks ◽

Driver Genes ◽

Complexity Measures ◽

Cellular Processes ◽

Graph Energy ◽

Energy Cutoff ◽

Cancer Networks ◽

Irreducible Graphs

Biological processes are based on molecular networks, which exhibit biological functions through interactions of genetic elements or proteins. This study presents a graph-based method to characterize molecular networks by decomposing the networks into directed multigraphs: network subgraphs. Spectral graph theory, reciprocity and complexity measures were used to quantify the network subgraphs. Graph energy, reciprocity and cyclomatic complexity can optimally specify network subgraphs with some degree of degeneracy. Seventy-one molecular networks were analyzed from three network types: cancer networks, signal transduction networks, and cellular processes. Molecular networks are built from a finite number of subgraph patterns and subgraphs with large graph energies are not present, which implies a graph energy cutoff. In addition, certain subgraph patterns are absent from the three network types. Thus, the Shannon entropy of the subgraph frequency distribution is not maximal. Furthermore, frequently-observed subgraphs are irreducible graphs. These novel findings warrant further investigation and may lead to important applications. Finally, we observed that cancer-related cellular processes are enriched with subgraph-associated driver genes. Our study provides a systematic approach for dissecting biological networks and supports the conclusion that there are organizational principles underlying molecular networks.

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