A Theorem of Kuratowski

Dirac, G. A.; Schuster, S. A.

doi:10.1016/s1385-7258(54)50043-0

Cited by 71 publications

(23 citation statements)

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“…We are now ready to prove Kuratowski's theorem. Our proof is based on that of Dirac and Schuster (1954 Proof We have already noted that the necessity follows from lemmas 9.10.1 and 9.10.2. We shall prove the sufficiency by contradiction.…”

Section: Graph Theory With Applicationsmentioning

confidence: 97%

Graph Theory with Applications

Bondy¹

1976

7,006

3,571

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Section: Graph Theory With Applicationsmentioning

confidence: 97%

Graph Theory with Applications

Bondy¹

1976

7,006

3,571

View full text Add to dashboard Cite

“…The idea of the construction of planar graphs is based on connecting the most correlated agents iteratively while constraining the resulting network to be embedded on a surface with genus g. In [42], authors briefly studied the special case for g D 0; i.e., the graph embedded on a sphere, and called it as the Planar Maximal Filter Graph (PMFG). PFMG can be seen as the topological triangulation of the sphere, henceforth it is only allowed to have three or four cliques [43].…”

Section: Graph Theorymentioning

confidence: 99%

Fractional virus epidemic model on financial networks

Balcı

2016

Open Mathematics

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Abstract:In this study, we present an epidemic model that characterizes the behavior of a financial network of globally operating stock markets. Since the long time series have a global memory effect, we represent our model by using the fractional calculus. This model operates on a network, where vertices are the stock markets and edges are constructed by the correlation distances. Thereafter, we find an analytical solution to commensurate system and use the well-known differential transform method to obtain the solution of incommensurate system of fractional differential equations. Our findings are confirmed and complemented by the data set of the relevant stock markets between 2006 and 2016. Rather than the hypothetical values, we use the Hurst Exponent of each time series to approximate the fraction size and graph theoretical concepts to obtain the variables.

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“…As this graph admits plane embedding (see Fig. 19), and by applying the Kuratowski theorem for infinite graphs 1 (see Dirac and Schuster, 1954): Fig. 18.…”

Section: Proofmentioning

confidence: 99%

An advance in infinite graph models for the analysis of transportation networks

Cera

Fedriani

2016

International Journal of Applied Mathematics and Computer Science

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This paper extends to infinite graphs the most general extremal issues, which are problems of determining the maximum number of edges of a graph not containing a given subgraph. It also relates the new results with the corresponding situations for the finite case. In particular, concepts from 'finite' graph theory, like the average degree and the extremal number, are generalized and computed for some specific cases. Finally, some applications of infinite graphs to the transportation of dangerous goods are presented; they involve the analysis of networks and percolation thresholds.

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A Theorem of Kuratowski

Cited by 71 publications

References 0 publications

Graph Theory with Applications

Graph Theory with Applications

Fractional virus epidemic model on financial networks

An advance in infinite graph models for the analysis of transportation networks

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