Extrema concerning asymmetric graphs

Quintas, Louis V.

doi:10.1016/s0021-9800(67)80018-8

Cited by 19 publications

(24 citation statements)

References 5 publications

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“…That is because with an increasing number of edges it becomes impossible to partition the set of nodes in a way that nodes within clusters are more connected to each other than to nodes in different clusters (for details see [28,30]). On the other hand, a graph is likely to become symmetric if more and more edges are added, which increases its density [31].…”

Section: Resultsmentioning

confidence: 99%

How Symmetric Are Real-World Graphs? A Large-Scale Study

Ball

Geyer-Schulz

2018

Symmetry

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Abstract:The analysis of symmetry is a main principle in natural sciences, especially physics. For network sciences, for example, in social sciences, computer science and data science, only a few small-scale studies of the symmetry of complex real-world graphs exist. Graph symmetry is a topic rooted in mathematics and is not yet well-received and applied in practice. This article underlines the importance of analyzing symmetry by showing the existence of symmetry in real-world graphs. An analysis of over 1500 graph datasets from the meta-repository networkrepository.com is carried out and a normalized version of the "network redundancy" measure is presented. It quantifies graph symmetry in terms of the number of orbits of the symmetry group from zero (no symmetries) to one (completely symmetric), and improves the recognition of asymmetric graphs. Over 70% of the analyzed graphs contain symmetries (i.e., graph automorphisms), independent of size and modularity. Therefore, we conclude that real-world graphs are likely to contain symmetries. This contribution is the first larger-scale study of symmetry in graphs and it shows the necessity of handling symmetry in data analysis: The existence of symmetries in graphs is the cause of two problems in graph clustering we are aware of, namely, the existence of multiple equivalent solutions with the same value of the clustering criterion and, secondly, the inability of all standard partition-comparison measures of cluster analysis to identify automorphic partitions as equivalent.

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Section: Resultsmentioning

confidence: 99%

How Symmetric Are Real-World Graphs? A Large-Scale Study

Ball

Geyer-Schulz

2018

Symmetry

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“…Otherwise, we proceed by induction on e. For e = e ( a , n) the statement follows from Ref. 6. Suppose that e(&, n) < e 5 n -2 and the statement holds for e-1.…”

Section: Identity Groupmentioning

confidence: 99%

“…For n = 6, E ( d , 6) = 9 [6] and $(3)=10.5. Thus we have shown E ( d , n ) Z~( "~' ) J -l for n r 6 .…”

Section: Identity Groupmentioning

confidence: 99%

An intermediate value theorem for graphs with given automorphism group

Hell

Quintas

1979

Journal of Graph Theory

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For a positive integer n and a finite group %, let the symbols e(%, n ) and €(%, n ) denote, respectively, the smallest and the greatest number of lines among all n-point graphs with automorphism group %. We say that the Intermediate Value Theorem (IVT) holds for % and n, if for each e satisfying e(%, n )~ e 5 €(%, n), there exists an n-point graph with group % and e lines. The main result of this paper states that for every group % the IVT holds for all sufficiently large n. We also prove that the IVT holds for the identity group and all n, and exhibit examples of groups for which the IVT fails to hold for small values of n.

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“…The value of