Введение
Граф, в котором все упорядоченные пары связанных узлов являются автоморфными. В математической области теории графов, граф G симметричен (или переходный по дуге), если для любых двух пар смежных вершин и в G существует автоморфизм
Graph in which all ordered pairs of linked nodes are automorphic
In the mathematical field of graph theory, a graph G is symmetric (or arc transitive) if, given any two pairs of adjacent vertices and of G, there is an automorphism
such that
and
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1 arc transitive
By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex transitive. Such graphs are called half transitive. The smallest connected half transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex transitive and edge transitive, rather than an arc transitive graph. Such a definition would include half transitive graphs, which are excluded under the definition above. A distance transitive graph is one where instead of considering pairs of adjacent vertices (i. e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs, and in 1988 (when Foster was 92 The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices (ten of these are also distance transitive; the exceptions are as indicated):
Vertices Diameter Girth Graph Notes4 1 3 The complete graph K4 distance transitive, 2 arc transitive6 2 4 The complete bipartite graph K3,3 distance transitive, 3 arc transitive8 3 4 The vertices and edges of the cube distance transitive, 2 arc transitive10 2 5 The Petersen graph distance transitive, 3 arc transitive14 3 6 The Heawood graph distance transitive, 4 arc transitive16 4 6 The Möbius–Kantor graph 2 arc transitive18 4 6 The Pappus graph distance transitive, 3 arc transitive20 5 5 The vertices and edges of the dodecahedron distance transitive, 2 arc transitive20 5 6 The Desargues graph distance transitive, 3 arc transitive24 4 6 The Nauru graph (the generalized Petersen graph G(12,5)) 2 arc transitive26 5 6 The F26A graph 1 arc transitive28 4 7 The Coxeter graph distance transitive, 3 arc transitive30 4 8 The Tutte–Coxeter graph distance transitive, 5 arc transitive
Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance transitive graphs.
In the mathematical field of graph theory, a graph G is symmetric (or arc transitive) if, given any two pairs of adjacent vertices and of G, there is an automorphism
such that
and
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1 arc transitive
By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex transitive. Such graphs are called half transitive. The smallest connected half transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex transitive and edge transitive, rather than an arc transitive graph. Such a definition would include half transitive graphs, which are excluded under the definition above. A distance transitive graph is one where instead of considering pairs of adjacent vertices (i. e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs, and in 1988 (when Foster was 92 The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices (ten of these are also distance transitive; the exceptions are as indicated):
Vertices Diameter Girth Graph Notes4 1 3 The complete graph K4 distance transitive, 2 arc transitive6 2 4 The complete bipartite graph K3,3 distance transitive, 3 arc transitive8 3 4 The vertices and edges of the cube distance transitive, 2 arc transitive10 2 5 The Petersen graph distance transitive, 3 arc transitive14 3 6 The Heawood graph distance transitive, 4 arc transitive16 4 6 The Möbius–Kantor graph 2 arc transitive18 4 6 The Pappus graph distance transitive, 3 arc transitive20 5 5 The vertices and edges of the dodecahedron distance transitive, 2 arc transitive20 5 6 The Desargues graph distance transitive, 3 arc transitive24 4 6 The Nauru graph (the generalized Petersen graph G(12,5)) 2 arc transitive26 5 6 The F26A graph 1 arc transitive28 4 7 The Coxeter graph distance transitive, 3 arc transitive30 4 8 The Tutte–Coxeter graph distance transitive, 5 arc transitive
Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance transitive graphs.
такой, что
Graph in which all ordered pairs of linked nodes are automorphic
In the mathematical field of graph theory, a graph G is symmetric (or arc transitive) if, given any two pairs of adjacent vertices and of G, there is an automorphism
such that
and
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1 arc transitive
By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex transitive. Such graphs are called half transitive. The smallest connected half transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex transitive and edge transitive, rather than an arc transitive graph. Such a definition would include half transitive graphs, which are excluded under the definition above. A distance transitive graph is one where instead of considering pairs of adjacent vertices (i. e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs, and in 1988 (when Foster was 92 The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices (ten of these are also distance transitive; the exceptions are as indicated):
Vertices Diameter Girth Graph Notes4 1 3 The complete graph K4 distance transitive, 2 arc transitive6 2 4 The complete bipartite graph K3,3 distance transitive, 3 arc transitive8 3 4 The vertices and edges of the cube distance transitive, 2 arc transitive10 2 5 The Petersen graph distance transitive, 3 arc transitive14 3 6 The Heawood graph distance transitive, 4 arc transitive16 4 6 The Möbius–Kantor graph 2 arc transitive18 4 6 The Pappus graph distance transitive, 3 arc transitive20 5 5 The vertices and edges of the dodecahedron distance transitive, 2 arc transitive20 5 6 The Desargues graph distance transitive, 3 arc transitive24 4 6 The Nauru graph (the generalized Petersen graph G(12,5)) 2 arc transitive26 5 6 The F26A graph 1 arc transitive28 4 7 The Coxeter graph distance transitive, 3 arc transitive30 4 8 The Tutte–Coxeter graph distance transitive, 5 arc transitive
Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance transitive graphs.
In the mathematical field of graph theory, a graph G is symmetric (or arc transitive) if, given any two pairs of adjacent vertices and of G, there is an automorphism
such that
and
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1 arc transitive
By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex transitive. Such graphs are called half transitive. The smallest connected half transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex transitive and edge transitive, rather than an arc transitive graph. Such a definition would include half transitive graphs, which are excluded under the definition above. A distance transitive graph is one where instead of considering pairs of adjacent vertices (i. e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs, and in 1988 (when Foster was 92 The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices (ten of these are also distance transitive; the exceptions are as indicated):
Vertices Diameter Girth Graph Notes4 1 3 The complete graph K4 distance transitive, 2 arc transitive6 2 4 The complete bipartite graph K3,3 distance transitive, 3 arc transitive8 3 4 The vertices and edges of the cube distance transitive, 2 arc transitive10 2 5 The Petersen graph distance transitive, 3 arc transitive14 3 6 The Heawood graph distance transitive, 4 arc transitive16 4 6 The Möbius–Kantor graph 2 arc transitive18 4 6 The Pappus graph distance transitive, 3 arc transitive20 5 5 The vertices and edges of the dodecahedron distance transitive, 2 arc transitive20 5 6 The Desargues graph distance transitive, 3 arc transitive24 4 6 The Nauru graph (the generalized Petersen graph G(12,5)) 2 arc transitive26 5 6 The F26A graph 1 arc transitive28 4 7 The Coxeter graph distance transitive, 3 arc transitive30 4 8 The Tutte–Coxeter graph distance transitive, 5 arc transitive
Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance transitive graphs.
и
Graph in which all ordered pairs of linked nodes are automorphic
In the mathematical field of graph theory, a graph G is symmetric (or arc transitive) if, given any two pairs of adjacent vertices and of G, there is an automorphism
such that
and
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1 arc transitive
By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex transitive. Such graphs are called half transitive. The smallest connected half transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex transitive and edge transitive, rather than an arc transitive graph. Such a definition would include half transitive graphs, which are excluded under the definition above. A distance transitive graph is one where instead of considering pairs of adjacent vertices (i. e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs, and in 1988 (when Foster was 92 The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices (ten of these are also distance transitive; the exceptions are as indicated):
Vertices Diameter Girth Graph Notes4 1 3 The complete graph K4 distance transitive, 2 arc transitive6 2 4 The complete bipartite graph K3,3 distance transitive, 3 arc transitive8 3 4 The vertices and edges of the cube distance transitive, 2 arc transitive10 2 5 The Petersen graph distance transitive, 3 arc transitive14 3 6 The Heawood graph distance transitive, 4 arc transitive16 4 6 The Möbius–Kantor graph 2 arc transitive18 4 6 The Pappus graph distance transitive, 3 arc transitive20 5 5 The vertices and edges of the dodecahedron distance transitive, 2 arc transitive20 5 6 The Desargues graph distance transitive, 3 arc transitive24 4 6 The Nauru graph (the generalized Petersen graph G(12,5)) 2 arc transitive26 5 6 The F26A graph 1 arc transitive28 4 7 The Coxeter graph distance transitive, 3 arc transitive30 4 8 The Tutte–Coxeter graph distance transitive, 5 arc transitive
Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance transitive graphs.
In the mathematical field of graph theory, a graph G is symmetric (or arc transitive) if, given any two pairs of adjacent vertices and of G, there is an automorphism
such that
and
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1 arc transitive
By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex transitive. Such graphs are called half transitive. The smallest connected half transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex transitive and edge transitive, rather than an arc transitive graph. Such a definition would include half transitive graphs, which are excluded under the definition above. A distance transitive graph is one where instead of considering pairs of adjacent vertices (i. e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs, and in 1988 (when Foster was 92 The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices (ten of these are also distance transitive; the exceptions are as indicated):
Vertices Diameter Girth Graph Notes4 1 3 The complete graph K4 distance transitive, 2 arc transitive6 2 4 The complete bipartite graph K3,3 distance transitive, 3 arc transitive8 3 4 The vertices and edges of the cube distance transitive, 2 arc transitive10 2 5 The Petersen graph distance transitive, 3 arc transitive14 3 6 The Heawood graph distance transitive, 4 arc transitive16 4 6 The Möbius–Kantor graph 2 arc transitive18 4 6 The Pappus graph distance transitive, 3 arc transitive20 5 5 The vertices and edges of the dodecahedron distance transitive, 2 arc transitive20 5 6 The Desargues graph distance transitive, 3 arc transitive24 4 6 The Nauru graph (the generalized Petersen graph G(12,5)) 2 arc transitive26 5 6 The F26A graph 1 arc transitive28 4 7 The Coxeter graph distance transitive, 3 arc transitive30 4 8 The Tutte–Coxeter graph distance transitive, 5 arc transitive
Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance transitive graphs.
Другими словами, граф симметричен, если его группа автоморфизмов действует транзитивно на упорядоченные пары смежных вершин (то есть на ребра, рассматриваемые как направленные). Такой граф иногда также называют 1-дуговым транзитивным.
Graph in which all ordered pairs of linked nodes are automorphic
In the mathematical field of graph theory, a graph G is symmetric (or arc transitive) if, given any two pairs of adjacent vertices and of G, there is an automorphism
such that
and
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1 arc transitive
By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex transitive. Such graphs are called half transitive. The smallest connected half transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex transitive and edge transitive, rather than an arc transitive graph. Such a definition would include half transitive graphs, which are excluded under the definition above. A distance transitive graph is one where instead of considering pairs of adjacent vertices (i. e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs, and in 1988 (when Foster was 92 The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices (ten of these are also distance transitive; the exceptions are as indicated):
Vertices Diameter Girth Graph Notes4 1 3 The complete graph K4 distance transitive, 2 arc transitive6 2 4 The complete bipartite graph K3,3 distance transitive, 3 arc transitive8 3 4 The vertices and edges of the cube distance transitive, 2 arc transitive10 2 5 The Petersen graph distance transitive, 3 arc transitive14 3 6 The Heawood graph distance transitive, 4 arc transitive16 4 6 The Möbius–Kantor graph 2 arc transitive18 4 6 The Pappus graph distance transitive, 3 arc transitive20 5 5 The vertices and edges of the dodecahedron distance transitive, 2 arc transitive20 5 6 The Desargues graph distance transitive, 3 arc transitive24 4 6 The Nauru graph (the generalized Petersen graph G(12,5)) 2 arc transitive26 5 6 The F26A graph 1 arc transitive28 4 7 The Coxeter graph distance transitive, 3 arc transitive30 4 8 The Tutte–Coxeter graph distance transitive, 5 arc transitive
Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance transitive graphs.
In the mathematical field of graph theory, a graph G is symmetric (or arc transitive) if, given any two pairs of adjacent vertices and of G, there is an automorphism
such that
and
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1 arc transitive
By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex transitive. Such graphs are called half transitive. The smallest connected half transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex transitive and edge transitive, rather than an arc transitive graph. Such a definition would include half transitive graphs, which are excluded under the definition above. A distance transitive graph is one where instead of considering pairs of adjacent vertices (i. e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs, and in 1988 (when Foster was 92 The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices (ten of these are also distance transitive; the exceptions are as indicated):
Vertices Diameter Girth Graph Notes4 1 3 The complete graph K4 distance transitive, 2 arc transitive6 2 4 The complete bipartite graph K3,3 distance transitive, 3 arc transitive8 3 4 The vertices and edges of the cube distance transitive, 2 arc transitive10 2 5 The Petersen graph distance transitive, 3 arc transitive14 3 6 The Heawood graph distance transitive, 4 arc transitive16 4 6 The Möbius–Kantor graph 2 arc transitive18 4 6 The Pappus graph distance transitive, 3 arc transitive20 5 5 The vertices and edges of the dodecahedron distance transitive, 2 arc transitive20 5 6 The Desargues graph distance transitive, 3 arc transitive24 4 6 The Nauru graph (the generalized Petersen graph G(12,5)) 2 arc transitive26 5 6 The F26A graph 1 arc transitive28 4 7 The Coxeter graph distance transitive, 3 arc transitive30 4 8 The Tutte–Coxeter graph distance transitive, 5 arc transitive
Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance transitive graphs.
По определению (не принимая во внимание и ), симметричный граф без изолированных вершин также должен быть вершинно транзитивным. Такие графы называются полутранзитивными. Наименьший связный полутранзитивный граф — это граф Холта, со степенью 4 и 27 вершинами. Некоторые авторы используют термин «симметричный граф» для обозначения графа, который является вершинно транзитивным и реберно транзитивным, а не дуговым транзитивным. Такое определение включает в себя полутранзитивные графы, которые исключены в соответствии с вышеуказанным определением. Граф, транзитивный по расстояниям, — это граф, в котором вместо рассмотрения пар смежных вершин (то есть вершин на расстоянии 1 друг от друга) определение охватывает две пары вершин, каждая из которых находится на одинаковом расстоянии друг от друга. Такие графы по определению автоматически симметричны. Перепись Фостера была начата в 1930-х годах Рональдом М. Фостером, когда он работал в Bell Labs, а в 1988 году (когда Фостеру было 92 года) были опубликованы первые результаты. Первые тринадцать пунктов в списке — кубические симметричные графы с до 30 вершин (десять из них также являются транзитивными по расстояниям; исключения указаны):
Graph in which all ordered pairs of linked nodes are automorphic
In the mathematical field of graph theory, a graph G is symmetric (or arc transitive) if, given any two pairs of adjacent vertices and of G, there is an automorphism
such that
and
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1 arc transitive
By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex transitive. Such graphs are called half transitive. The smallest connected half transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex transitive and edge transitive, rather than an arc transitive graph. Such a definition would include half transitive graphs, which are excluded under the definition above. A distance transitive graph is one where instead of considering pairs of adjacent vertices (i. e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs, and in 1988 (when Foster was 92 The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices (ten of these are also distance transitive; the exceptions are as indicated):
Vertices Diameter Girth Graph Notes4 1 3 The complete graph K4 distance transitive, 2 arc transitive6 2 4 The complete bipartite graph K3,3 distance transitive, 3 arc transitive8 3 4 The vertices and edges of the cube distance transitive, 2 arc transitive10 2 5 The Petersen graph distance transitive, 3 arc transitive14 3 6 The Heawood graph distance transitive, 4 arc transitive16 4 6 The Möbius–Kantor graph 2 arc transitive18 4 6 The Pappus graph distance transitive, 3 arc transitive20 5 5 The vertices and edges of the dodecahedron distance transitive, 2 arc transitive20 5 6 The Desargues graph distance transitive, 3 arc transitive24 4 6 The Nauru graph (the generalized Petersen graph G(12,5)) 2 arc transitive26 5 6 The F26A graph 1 arc transitive28 4 7 The Coxeter graph distance transitive, 3 arc transitive30 4 8 The Tutte–Coxeter graph distance transitive, 5 arc transitive
Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance transitive graphs.
In the mathematical field of graph theory, a graph G is symmetric (or arc transitive) if, given any two pairs of adjacent vertices and of G, there is an automorphism
such that
and
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1 arc transitive
By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex transitive. Such graphs are called half transitive. The smallest connected half transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex transitive and edge transitive, rather than an arc transitive graph. Such a definition would include half transitive graphs, which are excluded under the definition above. A distance transitive graph is one where instead of considering pairs of adjacent vertices (i. e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs, and in 1988 (when Foster was 92 The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices (ten of these are also distance transitive; the exceptions are as indicated):
Vertices Diameter Girth Graph Notes4 1 3 The complete graph K4 distance transitive, 2 arc transitive6 2 4 The complete bipartite graph K3,3 distance transitive, 3 arc transitive8 3 4 The vertices and edges of the cube distance transitive, 2 arc transitive10 2 5 The Petersen graph distance transitive, 3 arc transitive14 3 6 The Heawood graph distance transitive, 4 arc transitive16 4 6 The Möbius–Kantor graph 2 arc transitive18 4 6 The Pappus graph distance transitive, 3 arc transitive20 5 5 The vertices and edges of the dodecahedron distance transitive, 2 arc transitive20 5 6 The Desargues graph distance transitive, 3 arc transitive24 4 6 The Nauru graph (the generalized Petersen graph G(12,5)) 2 arc transitive26 5 6 The F26A graph 1 arc transitive28 4 7 The Coxeter graph distance transitive, 3 arc transitive30 4 8 The Tutte–Coxeter graph distance transitive, 5 arc transitive
Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance transitive graphs.
Вершин Диаметр Длина окружности Граф Примечания4 1 3 Полный граф K4 транзитивен по расстояниям, 2-дуговой транзитивен6 2 4 Полный двудольный граф K3,3 транзитивен по расстояниям, 3-дуговой транзитивен8 3 4 Вершины и ребра куба транзитивен по расстояниям, 2-дуговой транзитивен10 2 5 Граф Петерсена транзитивен по расстояниям, 3-дуговой транзитивен14 3 6 Граф Хивуда транзитивен по расстояниям, 4-дуговой транзитивен16 4 6 Граф Мёбиуса — Кантора 2-дуговой транзитивен18 4 6 Граф Паппа транзитивен по расстояниям, 3-дуговой транзитивен20 5 5 Вершины и ребра додекаэдра транзитивен по расстояниям, 2-дуговой транзитивен20 5 6 Граф Дезарга транзитивен по расстояниям, 3-дуговой транзитивен24 4 6 Граф Науру (обобщенный граф Петерсена G(12,5)) 2-дуговой транзитивен26 5 6 Граф F26A 1-дуговой транзитивен28 4 7 Граф Коксетера транзитивен по расстояниям, 3-дуговой транзитивен30 4 8 Граф Татта — Коксетера транзитивен по расстояниям, 5-дуговой транзитивен
Graph in which all ordered pairs of linked nodes are automorphic
In the mathematical field of graph theory, a graph G is symmetric (or arc transitive) if, given any two pairs of adjacent vertices and of G, there is an automorphism
such that
and
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1 arc transitive
By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex transitive. Such graphs are called half transitive. The smallest connected half transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex transitive and edge transitive, rather than an arc transitive graph. Such a definition would include half transitive graphs, which are excluded under the definition above. A distance transitive graph is one where instead of considering pairs of adjacent vertices (i. e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs, and in 1988 (when Foster was 92 The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices (ten of these are also distance transitive; the exceptions are as indicated):
Vertices Diameter Girth Graph Notes4 1 3 The complete graph K4 distance transitive, 2 arc transitive6 2 4 The complete bipartite graph K3,3 distance transitive, 3 arc transitive8 3 4 The vertices and edges of the cube distance transitive, 2 arc transitive10 2 5 The Petersen graph distance transitive, 3 arc transitive14 3 6 The Heawood graph distance transitive, 4 arc transitive16 4 6 The Möbius–Kantor graph 2 arc transitive18 4 6 The Pappus graph distance transitive, 3 arc transitive20 5 5 The vertices and edges of the dodecahedron distance transitive, 2 arc transitive20 5 6 The Desargues graph distance transitive, 3 arc transitive24 4 6 The Nauru graph (the generalized Petersen graph G(12,5)) 2 arc transitive26 5 6 The F26A graph 1 arc transitive28 4 7 The Coxeter graph distance transitive, 3 arc transitive30 4 8 The Tutte–Coxeter graph distance transitive, 5 arc transitive
Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance transitive graphs.
In the mathematical field of graph theory, a graph G is symmetric (or arc transitive) if, given any two pairs of adjacent vertices and of G, there is an automorphism
such that
and
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1 arc transitive
By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex transitive. Such graphs are called half transitive. The smallest connected half transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex transitive and edge transitive, rather than an arc transitive graph. Such a definition would include half transitive graphs, which are excluded under the definition above. A distance transitive graph is one where instead of considering pairs of adjacent vertices (i. e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs, and in 1988 (when Foster was 92 The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices (ten of these are also distance transitive; the exceptions are as indicated):
Vertices Diameter Girth Graph Notes4 1 3 The complete graph K4 distance transitive, 2 arc transitive6 2 4 The complete bipartite graph K3,3 distance transitive, 3 arc transitive8 3 4 The vertices and edges of the cube distance transitive, 2 arc transitive10 2 5 The Petersen graph distance transitive, 3 arc transitive14 3 6 The Heawood graph distance transitive, 4 arc transitive16 4 6 The Möbius–Kantor graph 2 arc transitive18 4 6 The Pappus graph distance transitive, 3 arc transitive20 5 5 The vertices and edges of the dodecahedron distance transitive, 2 arc transitive20 5 6 The Desargues graph distance transitive, 3 arc transitive24 4 6 The Nauru graph (the generalized Petersen graph G(12,5)) 2 arc transitive26 5 6 The F26A graph 1 arc transitive28 4 7 The Coxeter graph distance transitive, 3 arc transitive30 4 8 The Tutte–Coxeter graph distance transitive, 5 arc transitive
Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance transitive graphs.
Другие известные кубические симметричные графы — граф Дика, граф Фостера и граф Биггса — Смита. Десять графов, транзитивных по расстояниям, перечисленных выше, вместе с графом Фостера и графом Биггса — Смита, являются единственными кубическими графами, транзитивными по расстояниям.
Graph in which all ordered pairs of linked nodes are automorphic
In the mathematical field of graph theory, a graph G is symmetric (or arc transitive) if, given any two pairs of adjacent vertices and of G, there is an automorphism
such that
and
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1 arc transitive
By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex transitive. Such graphs are called half transitive. The smallest connected half transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex transitive and edge transitive, rather than an arc transitive graph. Such a definition would include half transitive graphs, which are excluded under the definition above. A distance transitive graph is one where instead of considering pairs of adjacent vertices (i. e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs, and in 1988 (when Foster was 92 The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices (ten of these are also distance transitive; the exceptions are as indicated):
Vertices Diameter Girth Graph Notes4 1 3 The complete graph K4 distance transitive, 2 arc transitive6 2 4 The complete bipartite graph K3,3 distance transitive, 3 arc transitive8 3 4 The vertices and edges of the cube distance transitive, 2 arc transitive10 2 5 The Petersen graph distance transitive, 3 arc transitive14 3 6 The Heawood graph distance transitive, 4 arc transitive16 4 6 The Möbius–Kantor graph 2 arc transitive18 4 6 The Pappus graph distance transitive, 3 arc transitive20 5 5 The vertices and edges of the dodecahedron distance transitive, 2 arc transitive20 5 6 The Desargues graph distance transitive, 3 arc transitive24 4 6 The Nauru graph (the generalized Petersen graph G(12,5)) 2 arc transitive26 5 6 The F26A graph 1 arc transitive28 4 7 The Coxeter graph distance transitive, 3 arc transitive30 4 8 The Tutte–Coxeter graph distance transitive, 5 arc transitive
Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance transitive graphs.
In the mathematical field of graph theory, a graph G is symmetric (or arc transitive) if, given any two pairs of adjacent vertices and of G, there is an automorphism
such that
and
In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1 arc transitive
By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex transitive. Such graphs are called half transitive. The smallest connected half transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex transitive and edge transitive, rather than an arc transitive graph. Such a definition would include half transitive graphs, which are excluded under the definition above. A distance transitive graph is one where instead of considering pairs of adjacent vertices (i. e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs, and in 1988 (when Foster was 92 The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices (ten of these are also distance transitive; the exceptions are as indicated):
Vertices Diameter Girth Graph Notes4 1 3 The complete graph K4 distance transitive, 2 arc transitive6 2 4 The complete bipartite graph K3,3 distance transitive, 3 arc transitive8 3 4 The vertices and edges of the cube distance transitive, 2 arc transitive10 2 5 The Petersen graph distance transitive, 3 arc transitive14 3 6 The Heawood graph distance transitive, 4 arc transitive16 4 6 The Möbius–Kantor graph 2 arc transitive18 4 6 The Pappus graph distance transitive, 3 arc transitive20 5 5 The vertices and edges of the dodecahedron distance transitive, 2 arc transitive20 5 6 The Desargues graph distance transitive, 3 arc transitive24 4 6 The Nauru graph (the generalized Petersen graph G(12,5)) 2 arc transitive26 5 6 The F26A graph 1 arc transitive28 4 7 The Coxeter graph distance transitive, 3 arc transitive30 4 8 The Tutte–Coxeter graph distance transitive, 5 arc transitive
Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance transitive graphs.
Свойства
Связность вершин симметричного графа всегда равна степени *d*. В отличие от этого, для вершинно-транзитивных графов в общем случае связность вершин ограничена снизу величиной 2(d + 1)/3. Вершинно-транзитивный граф степени *t*, равной 3 или более, имеет длину окружности не менее 2(*t* – 1). Однако не существует конечных вершинно-транзитивных графов степени 3 или более при *t* ≥ 8. В случае, когда степень равна ровно 3 (кубические симметричные графы), таких графов не существует при *t* ≥ 6.
The vertex connectivity of a symmetric graph is always equal to the degree d. In contrast, for vertex transitive graphs in general, the vertex connectivity is bounded below by 2(d + 1)/3. A t transitive graph of degree 3 or more has girth at least 2(t – 1). However, there are no finite t transitive graphs of degree 3 or more for t ≥ 8. In the case of the degree being exactly 3 (cubic symmetric graphs), there are none for t ≥ 6.