度

本词条由Ryan初步翻译

图1：A graph with a loop having vertices labeled by degree 内含自环按度标记的图

In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex, and in a multigraph, loops are counted twice.^[1] The degree of a vertex [math]\displaystyle{ v }[/math] is denoted [math]\displaystyle{ \deg(v) }[/math] or [math]\displaystyle{ \deg v }[/math]. The maximum degree of a graph [math]\displaystyle{ G }[/math], denoted by [math]\displaystyle{ \Delta(G) }[/math], and the minimum degree of a graph, denoted by [math]\displaystyle{ \delta(G) }[/math], are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.

In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex, and in a multigraph, loops are counted twice. The degree of a vertex [math]\displaystyle{ v }[/math] is denoted [math]\displaystyle{ \deg(v) }[/math] or [math]\displaystyle{ \deg v }[/math]. The maximum degree of a graph [math]\displaystyle{ G }[/math], denoted by [math]\displaystyle{ \Delta(G) }[/math], and the minimum degree of a graph, denoted by [math]\displaystyle{ \delta(G) }[/math], are the maximum and minimum degree of its vertices. In the multigraph on the right, the maximum degree is 5 and the minimum degree is 0.

在图论 Graph Theory中，图中顶点的度 Degree(或价)是入射到该顶点的边的数量。在多重图 Multigraph中，自环 Loops会被计算两次。顶点的度数可表示为[math]\displaystyle{ \deg(v) }[/math]或[math]\displaystyle{ \deg v }[/math]。一个图[math]\displaystyle{ G }[/math]的最大度值可表示为[math]\displaystyle{ \Delta(G) }[/math]，最小度值可表示为 [math]\displaystyle{ \delta(G) }[/math]。在右侧的多重图中，最大度值为5，最小度值为0。

In a regular graph, every vertex has the same degree, and so we can speak of the degree of the graph. A complete graph (denoted [math]\displaystyle{ K_n }[/math], where [math]\displaystyle{ n }[/math] is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, [math]\displaystyle{ n-1 }[/math].

In a regular graph, every vertex has the same degree, and so we can speak of the degree of the graph. A complete graph (denoted [math]\displaystyle{ K_n }[/math], where [math]\displaystyle{ n }[/math] is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum degree, [math]\displaystyle{ n-1 }[/math].

在正则图 Regular Graph中，每个顶点都具有相同的度数，因此我们可以将其称之为该图的度数。一个完全图 Complete Graph(表示为[math]\displaystyle{ K_n }[/math]，其中[math]\displaystyle{ n }[/math]是图中顶点的数目)是一种特殊的正则图，它所有顶点都有最大度值，[math]\displaystyle{ n-1 }[/math]。

Handshaking lemma 握手引理

握手引理

The degree sum formula states that, given a graph [math]\displaystyle{ G=(V, E) }[/math],

The degree sum formula states that, given a graph [math]\displaystyle{ G=(V, E) }[/math],

度和公式 Degree Sum Formula表明，任意给定一个图[math]\displaystyle{ G=(V, E) }[/math],都有

[math]\displaystyle{ \sum_{v \in V} \deg(v) = 2|E|\, . }[/math]

The formula implies that in any undirected graph, the number of vertices with odd degree is even. This statement (as well as the degree sum formula) is known as the handshaking lemma. The latter name comes from a popular mathematical problem, to prove that in any group of people the number of people who have shaken hands with an odd number of other people from the group is even.

此公式表明，在任何无向图中，拥有奇数度值的顶点的个数是偶数。这一阐释（以及度和公式)被称为握手引理 Handshaking Lemma。该名称来自一个有趣的数学问题，即求证无论该群体内有多少人，与奇数个人握过手的人数总是偶数。

 --趣木木（讨论）标题涉及到的专业名词 要么标注一下标题要么标注一下正文~

Degree sequence

度序列 Degree Sequence

图2：Two non-isomorphic graphs with the same degree sequence (3, 2, 2, 2, 2, 1, 1, 1). 两个具有相同度序列的非同构图

The degree sequence of an undirected graph is the non-increasing sequence of its vertex degrees;^[2] for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a graph invariant so isomorphic graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence.

The degree sequence of an undirected graph is the non-increasing sequence of its vertex degrees; for the above graph it is (5, 3, 3, 2, 2, 1, 0). The degree sequence is a graph invariant so isomorphic graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a graph; in some cases, non-isomorphic graphs have the same degree sequence.

无向图的度序列是指将其各顶点度值按非递增方式排序，对于上述图是(5,3,3,2,2,1,0)。度序列是图不变量 Graph Invariant，因此同构图 Non-isomorphic Graphs（两个图中顶点的度值都相同，但形状不同）具有相同的度序列。然而，度序列通常不能唯一地标识一个图，在某些情况下，非同构图也会具有相同的度序列。

 --CecileLi(讨论)  【审校】不确定：“无向图的度序列是指将其各顶点度值按非递增方式排序”一句中的“非递增”改为“递减/并不是按递增方式排序的”

The degree sequence problem is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a graphic or graphical sequence. As a consequence of the degree sum formula, any sequence with an odd sum, such as (3, 3, 1), cannot be realized as the degree sequence of a graph. The converse is also true: if a sequence has an even sum, it is the degree sequence of a multigraph. The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a matching, and fill out the remaining even degree counts by self-loops.

The degree sequence problem is the problem of finding some or all graphs with the degree sequence being a given non-increasing sequence of positive integers. (Trailing zeroes may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the graph.) A sequence which is the degree sequence of some graph, i.e. for which the degree sequence problem has a solution, is called a graphic or graphical sequence. As a consequence of the degree sum formula, any sequence with an odd sum, such as (3, 3, 1), cannot be realized as the degree sequence of a graph. The converse is also true: if a sequence has an even sum, it is the degree sequence of a multigraph. The construction of such a graph is straightforward: connect vertices with odd degrees in pairs by a matching, and fill out the remaining even degree counts by self-loops.

度序列问题 Degree Sequence Problem，是指寻找给定以非增方式排列的正整数的度序列的局部或全部图的问题。(序列尾部的零可能会被忽略，因为通过向图中添加适当数量的孤立顶点就可以轻松实现序列尾部不断加零。)一个序列是某个图的度序列，即一个度序列问题有解时，该序列称为图形序列。由于度和公式的存在，任何具有奇数和的序列，如(3,3,1) ，都不会是图的度序列。反之亦然: 如果一个序列和是偶数，它就是重图的度序列。我们可以轻易地构造一个图: 匹配奇数度值的顶点并成对连接起来，然后剩余的偶数度值顶点都连出一条边指向图形/起点本身。

The question of whether a given degree sequence can be realized by a simple graph is more challenging. This problem is also called graph realization problem and can either be solved by the Erdős–Gallai theorem or the Havel–Hakimi algorithm.

“一个给定的度序列是否可以用一个简单的图进行表示”是一个具有挑战性的问题。这个问题也称为图实现 Graph Realization Problem问题，但它可以用Erdős–Gallai定理 Erdős–Gallai Theoreme解决，也可以用Havel-Hakimi算法 Havel–Hakimi Algorithm解决。

The problem of finding or estimating the number of graphs with a given degree sequence is a problem from the field of graph enumeration.

基于给定度序列寻找或估计可能的图的个数，是图枚举 Graph Enumeration领域中的一个问题。

More generally, the degree sequence of a hypergraph is the non-increasing sequence of its vertex degrees. A sequence is [math]\displaystyle{ k }[/math]-graphic if it is the degree sequence of some [math]\displaystyle{ k }[/math]-uniform hypergraph. In particular, a [math]\displaystyle{ 2 }[/math]-graphic sequence is graphic. Deciding if a given sequence is [math]\displaystyle{ k }[/math]-graphic is doable in polynomial time for [math]\displaystyle{ k=2 }[/math] via the Erdős–Gallai theorem but is NP-complete for all [math]\displaystyle{ k\ge 3 }[/math] (Deza et al., 2018 ^[3]).

More generally, the degree sequence of a hypergraph is the non-increasing sequence of its vertex degrees. A sequence is [math]\displaystyle{ k }[/math]-graphic if it is the degree sequence of some [math]\displaystyle{ k }[/math]-uniform hypergraph. In particular, a [math]\displaystyle{ 2 }[/math]-graphic sequence is graphic. Deciding if a given sequence is [math]\displaystyle{ k }[/math]-graphic is doable in polynomial time for [math]\displaystyle{ k=2 }[/math] via the Erdős–Gallai theorem but is NP-complete for all [math]\displaystyle{ k\ge 3 }[/math] (Deza et al., 2018 ).

一般来说，超图 Hypergraph的度序列是其顶点度的非递增序列。如果一个序列是[math]\displaystyle{ k-uniform }[/math]超图的度序列，那么它即是[math]\displaystyle{ k-graphic }[/math]。特别地，[math]\displaystyle{ 2 }[/math]-graphic序列是图形。决定一个给定的序列是否是[math]\displaystyle{ k }[/math]-graphic可以在 [math]\displaystyle{ k=2 }[/math]的多项式时间内通过Erdős–Gallai定理实现，但是当[math]\displaystyle{ k\ge 3 }[/math]时该问题可转化为NP完全问题 NP-complete(Deza et al. ，2018)。

Special values

特殊值

图3：An undirected graph with leaf nodes 4, 5, 6, 7, 10, 11, and 12 一个具有叶节点的无定向图

A vertex with degree 0 is called an isolated vertex.

度数为0的顶点成为孤立顶点 Isolated Vertex

A vertex with degree 1 is called a leaf vertex or end vertex, and the edge incident with that vertex is called a pendant edge. In the graph on the right, {3,5} is a pendant edge. This terminology is common in the study of trees in graph theory and especially trees as data structures.

度数为1的顶点称为叶顶点或尾顶点，该顶点的入射边称为悬挂边 Pendant Edge。在右侧的图中，{3,5}就是一个悬挂边。在图论中，该术语主要在研究树 Tree时使用，特别是具有树形结构的数据。

A vertex with degree n − 1 in a graph on n vertices is called a dominating vertex.

在有n个顶点的图中，度数为n-1的顶点叫作主导顶点 Dominating Vertex

Global properties

全局属性

If each vertex of the graph has the same degree k the graph is called a k-regular graph and the graph itself is said to have degree k. Similarly, a bipartite graph in which every two vertices on the same side of the bipartition as each other have the same degree is called a biregular graph.

如果一个图中的所有顶点的度值都为k，该图被称为k-正则图 k-regular graph，该图的度值也为k。同样的，同侧每两个顶点都具有相同度值的二分图叫作二分正则图 Biregular Graph。

An undirected, connected graph has an Eulerian path if and only if it has either 0 or 2 vertices of odd degree. If it has 0 vertices of odd degree, the Eulerian path is an Eulerian circuit.

当且仅当具有0或2个奇数度的顶点时，无向连通图才具有欧拉路径 Eulerian Path。如果它具有0个奇数度的顶点，则欧拉路径为欧拉回路 Eulerian Circuit。

A directed graph is a pseudoforest if and only if every vertex has outdegree at most 1. A functional graph is a special case of a pseudoforest in which every vertex has outdegree exactly 1.

有向图是当且仅当每个顶点的度值最大为1时才是伪森林 Pseudoforest。功能图 Functional Graph是伪森林的特例，其中每个顶点的度数都恰好为1。

By Brooks' theorem, any graph other than a clique or an odd cycle has chromatic number at most Δ, and by Vizing's theorem any graph has chromatic index at most Δ + 1.

根据布鲁克斯定理，除团簇或奇数循环外，任何图的色度数最大为Δ，而根据维辛定理，任何图的色度指数最大为Δ+ 1。

A k-degenerate graph is a graph in which each subgraph has a vertex of degree at most k.

k简并图 K-Degenerate Graph 是其中每个子图最多具有度值为k的顶点的图。

Notes

↑ Diestel p.5
↑ Diestel p.216
↑ Deza, Antoine; Levin, Asaf; Meesum, Syed M.; Onn, Shmuel (January 2018). "Optimization over Degree Sequences". SIAM Journal on Discrete Mathematics (in English). 32 (3): 2067–2079. arXiv:1706.03951. doi:10.1137/17M1134482. ISSN 0895-4801.

References

Diestel, Reinhard (2005), Graph Theory (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-26183-4.

Erdős, P.; Gallai, T. (1960), "Gráfok előírt fokszámú pontokkal" (PDF), Matematikai Lapok (in Hungarian), 11: 264–274{{citation}}: CS1 maint: unrecognized language (link).

Havel, Václav (1955

1955年), [http://eudml.org/doc/19050

Http://eudml.org/doc/19050 "关于有限图存在性的一个注记"], Časopis pro pěstování matematiky (in Czech), 80

80: 477–480 {{citation}}: Check |url= value (help); Check date values in: |year= (help); Unknown parameter |页数= ignored (help); line feed character in |url= at position 27 (help); line feed character in |volume= at position 3 (help); line feed character in |year= at position 5 (help)CS1 maint: unrecognized language (link)

}}

Hakimi, S.l. (1962), "一组整数作为线性图顶点度的可实现性。一", Journal of the Society for Industrial and Applied Mathematics 工业与应用数学学会杂志, 10

10: 496–506, MR [//www.ams.org/mathscinet-getitem?mr=0148049%0A%0A0148049%E5%85%88%E7%94%9F 0148049 0148049先生] {{citation}}: Check |mr= value (help); Unknown parameter |页数= ignored (help); line feed character in |journal= at position 62 (help); line feed character in |mr= at position 8 (help); line feed character in |volume= at position 3 (help).

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1 = 杰拉德; Hoogeveen, Han (1991), "整数序列为图形的七个条件", Journal of Graph Theory, 15

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2): 223–231, doi:[//doi.org/10.1002%2Fjgt.3190150209%0A%0A10.1002%2Fjgt.%203190150209 10.1002/jgt.3190150209 10.1002/jgt. 3190150209], MR [//www.ams.org/mathscinet-getitem?mr=1106533%0A%0A1106533%E5%85%88%E7%94%9F 1106533 1106533先生] {{citation}}: Check |doi= value (help); Check |mr= value (help); Unknown parameter |Journal= ignored (|journal= suggested) (help); line feed character in |doi= at position 23 (help); line feed character in |first1= at position 7 (help); line feed character in |issue= at position 2 (help); line feed character in |mr= at position 8 (help); line feed character in |volume= at position 3 (help).

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1991}}.

Category:Graph theory

范畴: 图论

This page was moved from wikipedia:en:Degree (graph theory). Its edit history can be viewed at 度/edithistory

[1] Diestel p.5

[2] Diestel p.216

[3] Deza, Antoine; Levin, Asaf; Meesum, Syed M.; Onn, Shmuel (January 2018). "Optimization over Degree Sequences". SIAM Journal on Discrete Mathematics (in English). 32 (3): 2067–2079. arXiv:1706.03951. doi:10.1137/17M1134482. ISSN 0895-4801.

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