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| In [[graph theory]], a discipline within [[mathematics]], the '''frequency partition''' of a graph ([[simple graph]]) is a [[partition (number theory)|partition]] of its [[vertex (graph theory)|vertices]] grouped by their degree.For example, the [[degree sequence]] of the left-hand graph below is (3, 3, 3, 2, 2, 1) and its frequency partition is 6 = 3 + 2 + 1. This indicates that it has 3 vertices with some degree, 2 vertices with some other degree, and 1 vertex with a third degree. The degree sequence of the [[bipartite graph]] in the middle below is (3, 2, 2, 2, 2, 2, 1, 1, 1) and its frequency partition is 9 = 5 + 3 + 1. The degree sequence of the right-hand graph below is (3, 3, 3, 3, 3, 3, 2) and its frequency partition is 7 = 6 + 1.
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| <gallery>
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| Image:6n-graf.svg|A graph with frequency partition 6 = 3 + 2 + 1.
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| Image:Simple-bipartite-graph.svg|A bipartite graph with frequency partition 9 = 5 + 3 + 1.
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| Image:Nonplanar no subgraph K 3 3.svg|A graph with frequency partition 7 = 6 + 1.
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| </gallery>
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| In general, there are many non-isomorphic graphs with a given frequency partition. A graph and its [[Complement graph|complement]] have the same frequency partition. For any partition ''p'' = ''f''<sub>1</sub> + ''f''<sub>2</sub> + ... + ''f''<sub>''k''</sub> of an integer ''p'' > 1, other than ''p'' = 1 + 1 + 1 + ... + 1, there is at least one (connected) simple graph having this partition as its frequency partition.<ref>{{Citation |last=Chinn |first=P. Z. |authorlink=P. Z. Chinn |year=1971 |title=The frequency partition of a graph. Recent Trends in Graph Theory |work=Lecture Notes in Mathematics |volume=186 |pages=69–70 |publisher=Springer-Verlag |location=Berlin }}</ref>
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| Frequency partitions of various graph families are completely identifieds; frequency partitions of many families of graphs are not identified.
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| ==Frequency partitions of Eulerian graphs==
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| For a frequency [[Partition (number theory)|partition]] ''p'' = ''f''<sub>1</sub> + ''f''<sub>2</sub> + ... + ''f''<sub>''k''</sub> of an integer ''p'' > 1, its [[Degree (graph theory)|graphic degree sequence]] is denoted as ((d<sub>1</sub>)<sup>f<sub>1</sub></sup>,(d<sub>2</sub>)<sup>f<sub>2</sub></sup>, (d<sub>3</sub>)<sup>f<sub>3</sub></sup>, ..., (d<sub>k</sub>) <sup>f<sub>k</sub></sup>) where degrees d<sub>i</sub>'s are different and ''f''<sub>i</sub> ≥ ''f''<sub>j</sub> for ''i'' < ''j''.
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| Bhat-Nayak ''et al.'' (1979) showed that a partition of p with k parts, k ≤ integral part of <math>(p-1)/2</math> is a frequency partition<ref>{{Citation |last=Bhat-Nayak |first=Vasanti N. |last2=Naik |first2=Ranjan N. |last3=Rao |first3=S. B. |lastauthoramp=yes |year=1979 |chapter=Characterization of Frequency Partitions of Eulerian Graphs |title=ISI Lecture Notes |editor-first=A. R. |editor-last=Rao |work=Proc. Symposium on graph Theory |volume=4 |publisher=The MacMillan comp. of India }}. Also in Lecture Notes in Mathematics, Combinatorics and Graph Theory, [[Springer-Verlag]], New York, Vol. 885 (1980), p 500. </ref> of a Eulerian graph and conversely.
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| ==Frequency partition of trees, Hamiltonian graphs, tournaments and hypegraphs==
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| The frequency partitions of families of graphs such as [[tree]]s,<ref>{{Citation |last=Rao |first=T. M. |lastauthoramp= |year=1974 |title=Frequency sequences in Graphs |journal=[[Journal of Combinatorial Theory]] B |volume=17 |issue= |pages=19–21 |url= |issn= }}</ref> [[Hamiltonian graph]]s<ref>*{{Citation |last=Bhat-Nayak |first=Vasanti N. |authorlink=Vasanti N. Bhat-Nayak |last2=Naik |first2=Ranjan N. |last3=Rao |first3=S. B. |authorlink3=S. B. Rao |lastauthoramp=yes |year=1977 |title=Frequency partitions: forcibly pancyclic and forcibly nonhamiltonian degree sequences |journal=[[Discrete Mathematics (journal)|Discrete Mathematics]] |volume=20 |issue= |pages=93–102 |url= |issn= }}
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| </ref> [[directed graph]]s and [[Tournament (graph theory)|tournaments]]<ref>{{Citation |last=Alspach |first=B. |last2=Reid |first2=K. B. |authorlink2=K. B. Reid |lastauthoramp=yes |year=1978 |title=Degree Frequencies in Digraphs and Tournaments |journal=Journal of Graph Theory |volume=2 |issue= |pages=241–249 |doi=10.1002/jgt.3190020307 }}</ref> and to k-uniform [[hypergraph]]s.<ref>{{Citation |last=Bhat-Nayak |first=V. N. |last2=Naik |first2=R. N. |lastauthoramp=yes |year=1985 |title=Frequency partitions of k-uniform hypergraphs |journal=Utilitas Math. |volume=28 |issue= |pages=99–104 |url= |issn= }}</ref> have been characterized.
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| ==Unsolved problems in frequency partitions==
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| The frequency partitions of the following families of graphs have not yet been characterized:
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| * [[Line graph]]s
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| * [[Bipartite graph]]s <ref>[[S. B. Rao]], A survey of the theory of potentially p-graphic and forcibly p-graphic sequences, in: S. B. Rao edited., Combinatorics and Graph Theory Lecture Notes in Math., Vol. 885 (Springer, Berlin, 1981), 417-440 </ref>
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| ==References==
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| {{reflist}}
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| ==External section==
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| *{{citation |last=Berge |first=C. |authorlink=Claude Berge |chapter= |title=Hypergraphs, Combinatorics of Finite sets |year=1989 |publisher=North-Holland |location=Amsterdam |isbn=0-444-87489-5 }}
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| [[Category:Graph theory]]
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I'm Otis and I live with my husband and our three children in Albersroda, in the ST south area. My hobbies are RC cars, Videophilia (Home theater) and Meteorology.
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