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VertexInDegree[g]

グラフ g 中のすべての頂点の頂点入次数のリストを与える．

VertexInDegree[g,v]

頂点 v の頂点入次数を与える．

VertexInDegree[{vw,…},…]

規則 vw を使ってグラフ g を指定する．

詳細
Details and Options Details and Options
例題  
例  
スコープ  
アプリケーション  
特性と関係  
関連項目
関連するガイド
履歴
引用書式

VertexInDegree

VertexInDegree[g]

グラフ g 中のすべての頂点の頂点入次数のリストを与える．

VertexInDegree[g,v]

頂点 v の頂点入次数を与える．

VertexInDegree[{vw,…},…]

規則 vw を使ってグラフ g を指定する．

詳細

  • 頂点 v の頂点入次数とは v に入る有向辺の数である．
  • 無向グラフ g の辺は入辺かつ出辺であるとみなされる． »

例題

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例  (2)

各頂点の入次数を求める：

Wolfram Language code: VertexInDegree[[image]]

指定された頂点の入次数を求める：

Wolfram Language code: VertexInDegree[[image], 2]

スコープ  (6)

VertexInDegreeは有向グラフに使うことができる：

Wolfram Language code: VertexInDegree[[image]]

無向グラフ：

Wolfram Language code: VertexInDegree[[image]]

多重グラフ：

Wolfram Language code: VertexInDegree[[image]]

頂点の頂点入次数：

Wolfram Language code: VertexInDegree[[image], 5]

規則を使ってグラフを指定する：

Wolfram Language code: VertexInDegree[{1 -> 2, 2 -> 3, 3 -> 1, 3 -> 4}]

VertexInDegreeは大きいグラフに使うことができる：

Wolfram Language code: g = GridGraph[{10, 10, 10, 10}, DirectedEdges -> True];
Wolfram Language code: VertexInDegree[g]//Timing//Short

アプリケーション  (4)

CycleGraphを含む有向グラフの頂点入次数をハイライトする：

Wolfram Language code: HighlightVertexInDegree[g_, vd_] := HighlightGraph[g, Table[Style[VertexList[g][[i]], ColorData["TemperatureMap"][vd[[i]] / Max[vd]]], {i, VertexCount[g]}]];
Wolfram Language code: g = CycleGraph[8, DirectedEdges -> True, VertexSize -> Large];
Wolfram Language code: vd = VertexInDegree[g];
Wolfram Language code: HighlightVertexInDegree[g, vd]

StarGraph：

Wolfram Language code: g = StarGraph[10, DirectedEdges -> True, VertexSize -> Large];
Wolfram Language code: vd = VertexInDegree[g];
Wolfram Language code: HighlightVertexInDegree[g, vd]

GridGraph：

Wolfram Language code: g = GridGraph[{10, 10}, DirectedEdges -> True, VertexSize -> Large];
Wolfram Language code: vd = VertexInDegree[g];
Wolfram Language code: HighlightVertexInDegree[g, vd]

CompleteKaryTree：

Wolfram Language code: g = CompleteKaryTree[3, 3, DirectedEdges -> True, VertexSize -> Large];
Wolfram Language code: vd = VertexInDegree[g];
Wolfram Language code: HighlightVertexInDegree[g, vd]

PathGraph：

Wolfram Language code: g = PathGraph[Range[20], DirectedEdges -> True, VertexSize -> Large];
Wolfram Language code: vd = VertexInDegree[g];
Wolfram Language code: HighlightVertexInDegree[g, vd]

RandomGraph：

Wolfram Language code: g = RandomGraph[BarabasiAlbertGraphDistribution[50, 5], VertexSize -> {"Scaled", 0.03}, BaseStyle -> EdgeForm[None]];
Wolfram Language code: vd = VertexInDegree[g];
Wolfram Language code: HighlightVertexInDegree[g, vd]

BernoulliGraphDistribution[n,p]の入次数ヒストグラムを示す：

Wolfram Language code: g = RandomGraph[BernoulliGraphDistribution[10^4, 0.01], DirectedEdges -> True];
Wolfram Language code: Histogram[VertexInDegree[g], Automatic, "PDF"]

入次数の分布はBinomialDistribution[n-1,p]に従う：

Wolfram Language code: DiscretePlot[PDF[BinomialDistribution[10^4 - 1, 0.01], k], {k, 70, 130}]

PriceGraphDistributionの頂点入次数の分布はベキ法則に従う：

Wolfram Language code: g = RandomGraph[PriceGraphDistribution[10^3, 3, 1]];
Wolfram Language code: Histogram[VertexInDegree[g], {"Log", 10}, "LogProbability"]

動物と昆虫が何を食べるかを辺で示す食物連鎖図を作る：

Wolfram Language code: web = {"Aphid""Ladybug", "Beetle""Mouse", "Caterpillar""Beetle", "Caterpillar""Mouse", "Caterpillar""Towhee", "Grasshopper""Mouse", "Grasshopper""Towhee", "Ladybug""Towhee", "Louse""Mouse", "Mosquito""Dragonfly", "Mosquito""Towhee", "Mouse""Owl", "Sunflower""Aphid", "Sunflower""Caterpillar", "Sunflower""Grasshopper", "Sunflower""Louse", "Towhee""Owl"};
Wolfram Language code: shapes = {"Sunflower" -> [image], "Aphid" -> [image], "Ladybug" -> [image], "Towhee" -> [image], "Owl" -> [image], "Grasshopper" -> [image], "Louse" -> [image], "Caterpillar" -> [image], "Beetle" -> [image], "Mosquito" -> [image], "Dragonfly" -> [image], "Mouse" -> [image]};
Wolfram Language code: g = Graph[web];

入次数は捕食動物と被食動物の数に対応する：

Wolfram Language code: Graph[EdgeList[g], VertexShape -> shapes, VertexLabels -> Table[i -> VertexInDegree[g, i], {i, VertexList[g]}], VertexSize -> 0.7]

入次数が0の種は基礎種あるいは生産者と呼ばれる：

Wolfram Language code: VertexInDegree[g]
Wolfram Language code: Pick[VertexList[g], %, 0]

特性と関係  (10)

無向グラフの入次数は各頂点に接続する辺の数である：

Wolfram Language code: Graph[{12, 23, 31}, VertexLabels -> "Name", ImagePadding -> 10]
Wolfram Language code: VertexInDegree[%]

自己ループは2回数えられる：

Wolfram Language code: Graph[{12, 23, 31, 33}, VertexLabels -> "Name", ImagePadding -> 10]
Wolfram Language code: VertexInDegree[%]

無向グラフは各辺が入辺かつ出辺である有向グラフに相当する：

Wolfram Language code: {g = Graph[{12, 23, 31}], gd = Graph[Flatten[EdgeList[g] /. {u_v_ :> {uv, vu}}]]}
Wolfram Language code: {VertexInDegree[g], VertexInDegree[gd]}

無向グラフでは，頂点入次数と出次数は頂点次数に等しい：

Wolfram Language code: WheelGraph[6]
Wolfram Language code: VertexInDegree[%] == VertexOutDegree[%] == VertexDegree[%]

有向グラフでは，頂点入次数と出次数の総和が頂点次数である：

Wolfram Language code: g = Graph[{12, 23, 31, 24}];

頂点次数，入次数，出次数を，頂点の前，上，下に記す：

Wolfram Language code: Graph[EdgeList[g], VertexLabels -> Table[i -> Placed[{VertexDegree[g][[i]], VertexInDegree[g][[i]], VertexOutDegree[g][[i]]}, {Before, Above, Below}], {i, VertexCount[g]}], VertexSize -> Small, ImagePadding -> 10]
Wolfram Language code: VertexDegree[g] == VertexInDegree[g] + VertexOutDegree[g]

無向グラフの全頂点の入次数の総和は辺の数の二倍である：

Wolfram Language code: CycleGraph[4]
Wolfram Language code: Total[VertexInDegree[%]] == 2EdgeCount[%]

有向グラフの全頂点の入次数の総和は辺の数に等しい：

Wolfram Language code: Total[VertexInDegree[[image]]] == EdgeCount[[image]]

無向グラフの頂点入次数は隣接行列から得られる：

Wolfram Language code: g = Graph[{11, 12, 23, 31, 34}, VertexLabels -> "Name", ImagePadding -> 10]
Wolfram Language code: (a = AdjacencyMatrix[g])//MatrixForm
Wolfram Language code: VertexInDegree[g]
Wolfram Language code: Total[a] + Diagonal[a]

有向グラフの頂点入次数は隣接行列から得られる：

Wolfram Language code: g = [image];
Wolfram Language code: (a = AdjacencyMatrix[g])//MatrixForm
Wolfram Language code: VertexInDegree[g]
Wolfram Language code: Total[a]

無向グラフの頂点の入力次数は結合行列から得ることができる：

Wolfram Language code: g = Graph[{11, 12, 23, 31, 34}, VertexLabels -> "Name", ImagePadding -> 10]
Wolfram Language code: VertexInDegree[g]
Wolfram Language code: (i = IncidenceMatrix[g])//MatrixForm
Wolfram Language code: Total[i]

全頂点の入次数と出次数が等しいときかつそのときに限り連結有向グラフはユークリッドグラフである：

Wolfram Language code: g = [image];
Wolfram Language code: VertexInDegree[g] == VertexOutDegree[g]
Wolfram Language code: {ConnectedGraphQ[g], EulerianGraphQ[g]}
Wolfram Research (2010), VertexInDegree, Wolfram言語関数， https://reference.wolfram.com/language/ref/VertexInDegree.html （2015年に更新）.

テキスト

Wolfram Research (2010), VertexInDegree, Wolfram言語関数， https://reference.wolfram.com/language/ref/VertexInDegree.html （2015年に更新）.

CMS

Wolfram Language. 2010. "VertexInDegree." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/VertexInDegree.html.

APA

Wolfram Language. (2010). VertexInDegree. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/VertexInDegree.html

BibTeX

@misc{reference.wolfram_2026_vertexindegree, author="Wolfram Research", title="{VertexInDegree}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/VertexInDegree.html}", note=[Accessed: 09-November-2026]}

BibLaTeX

@online{reference.wolfram_2026_vertexindegree, organization={Wolfram Research}, title={VertexInDegree}, year={2015}, url={https://reference.wolfram.com/language/ref/VertexInDegree.html}, note=[Accessed: 09-November-2026]}