Кіріспе
График, онда байланыстырылған түйіндердің реттелген жұптарының барлығы автоматты түрде өзгереді.
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.
Граф теориясының математикалық саласында, G графигі симметриялық (немесе доғалық транзитивті) деп аталады, егер G графигіндегі кез келген екі іргелес түйін жұбы үшін, автоматты түрде
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.
болатын түрлендіру бар болса, мұнда
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.
және
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.
Басқаша айтқанда, граф симметриялық болады, егер оның автоматтылық тобы іргелес түйіндердің реттелген жұптарына (яғни бағыты бар жиектерге) транзитивті әрекет етсе. Мұндай граф кейде 1-доғалық транзитивті деп те аталады.
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.
Анықтама бойынша (және ескерместен), оқшауланған түйіндері жоқ симметриялық граф сонымен қатар түйіндық транзитивті болуы керек. Мұндай графтар жартылай транзитивті деп аталады. Ең кіші байланысты жартылай транзитивті граф – Холт графигі, 4 дәрежелі және 27 түйіні бар. Қызығы, кейбір авторлар «симметриялық граф» терминін доғалық транзитивті графқа қарағанда, түйіндық және жиектік транзитивті графты білдіру үшін қолданады. Мұндай анықтама жоғарыдағы анықтама бойынша шығарылмаған жартылай транзитивті графтарды қамтиды. Арақашықтық транзитивті граф – бұл граф, онда іргелес түйіндер жұбын (яғни 1 қашықтықтағы түйіндерді) қарастырудың орнына, анықтама бірдей қашықтықтағы екі түйін жұбын қамтиды. Мұндай графтар анықтама бойынша автоматты түрде симметриялық болады.
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.
Фостер санағын 1930 жылдары Рональд М. Фостер Bell Labs қызметкері болған кезде бастады, ал 1988 жылы (Фостер 92 жасқа толғанда) тізімдегі алғашқы он үш элемент 30 түйіні бар кубтық симметриялық графтар болып табылады (олардың оны арақашықтық транзитивті; ерекшеліктері көрсетілген):
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.
Түйіндер Диаметрі Айналым График Ескертулер
4 1 3 Толық граф K4 арақашықтық транзитивті, 2-доғалық транзитивті
6 2 4 Толық екі бөлікті граф K3,3 арақашықтық транзитивті, 3-доғалық транзитивті
8 3 4 Кубтың түйіндері мен жиектері арақашықтық транзитивті, 2-доғалық транзитивті
10 2 5 Петерсен графигі арақашықтық транзитивті, 3-доғалық транзитивті
14 3 6 Геавуд графигі арақашықтық транзитивті, 4-доғалық транзитивті
16 4 6 Мёбиус–Кантор графигі 2-доғалық транзитивті
18 4 6 Паппус графигі арақашықтық транзитивті, 3-доғалық транзитивті
20 5 5 Додекаэдрдың түйіндері мен жиектері арақашықтық транзитивті, 2-доғалық транзитивті
20 5 6 Дезарг графигі арақашықтық транзитивті, 3-доғалық транзитивті
24 4 6 Науру графигі (жалпыланған Петерсен графигі G(12,5)) 2-доғалық транзитивті
26 5 6 F26A графигі 1-доғалық транзитивті
28 4 7 Коксетер графигі арақашықтық транзитивті, 3-доғалық транзитивті
30 4 8 Татт–Коксетер графигі арақашықтық транзитивті, 5-доғалық транзитивті
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.
Басқа белгілі кубтық симметриялық графтар – Дик граф, Фостер граф және Биггс–Смит граф. Жоғарыда тізілген он арақашықтық транзитивті графтар, Фостер графигімен және Биггс–Смит графигімен бірге, тек кубтық арақашықтық транзитивті графтар болып табылады.
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.
Қасиеттері
Симметриялық графтың төбелік байланыстылығы әрқашан d дәрежесіне тең болады. Керісінше, төбелік транзитивті графтар үшін, жалпы алғанда, төбелік байланыстылығы 2(d + 1)/3-тен кем болмайды. 3 немесе одан жоғары дәрежелі t транзитивті графтың кемінде 2(t – 1) ұзындығы бар. Дегенмен, t ≥ 8 үшін 3 немесе одан жоғары дәрежелі шекті t транзитивті графтар жоқ. Дәрежесі нақты 3 болғанда (кубтық симметриялық графтар), t ≥ 6 үшін мұндай графтар жоқ.
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.