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To find the number of vertices in an octagonal pyramid using a graph, you can represent the pyramid as a 3D shape with vertices, edges, and faces. An octagonal pyramid has 8 vertices, one at the top (apex) and 8 at the base. You can also draw a graph with each vertex representing a corner of the pyramid and each edge representing a line connecting two vertices. By counting the number of vertices in the graph representation, you can determine that an octagonal pyramid has a total of 9 vertices.
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What is the largest number of vertices in a graph with 35 edges if all vertices are of degree at least 3?
11......
What is the automorphism group of a complete graph?
The automorphism group of a complete graph ( K_n ) (where ( n ) is the number of vertices) is the symmetric group ( S_n ). This is because any permutation of the vertices of ( K_n ) results in an isomorphic graph, as all vertices are equivalent in a complete graph. Therefore, the automorphism group consists of all possible ways to rearrange the vertices, corresponding to the ( n! ) permutations of the ( n ) vertices.
What is the largest number of vertices in a graph with 35 edges if all vertices are?
36 vertices if all of them are or order two except one at each end.
How can you show that every Hamiltonian cubic graph is 3-edge-colorable?
A cubic graph must have an even number of vertices. Then, a Hamilton cycle (visiting all vertices) must have an even number of vertices and also an even number of edges. Alternatively color this edges red and blue, and the remaining edges green.
What is the least possible number of vertices in a graph with 35 edges and all vertices of degree 3?
In a graph where all vertices have a degree of 3, the sum of the degrees of all vertices is equal to twice the number of edges. Therefore, if there are ( n ) vertices, the equation is ( 3n = 2 \times 35 = 70 ). Solving for ( n ) gives ( n = \frac{70}{3} ), which is approximately 23.33. Since ( n ) must be an integer, the least possible number of vertices is 24.
What is the cycle size of the given graph?
The cycle size of a graph is the number of vertices in the smallest cycle in the graph.
What is the largest number of vertices in a graph with 35 edges if all vertices are of degree at least 3?
11......
How can we determine the number of cycles in a graph?
To determine the number of cycles in a graph, you can use the concept of Euler's formula, which states that for a connected graph with V vertices, E edges, and F faces, the formula is V - E F 2. By calculating the number of vertices, edges, and faces in the graph, you can determine the number of cycles present.
How many minimum edges in a Cyclic graph with n vertices?
The term "cyclic graph" is not well-defined. If you mean a graph that is not acyclic, then the answer is 3. That would be the union of a complete graph on 3 vertices and any number of isolated vertices. If you mean a graph that is (isomorphic to) a cycle, then the answer is n. If you are really asking the maximum number of edges, then that would be the triangle numbers such as n (n-1) /2.
What is the automorphism group of a complete graph?
The automorphism group of a complete graph ( K_n ) (where ( n ) is the number of vertices) is the symmetric group ( S_n ). This is because any permutation of the vertices of ( K_n ) results in an isomorphic graph, as all vertices are equivalent in a complete graph. Therefore, the automorphism group consists of all possible ways to rearrange the vertices, corresponding to the ( n! ) permutations of the ( n ) vertices.
What is the significance of a minimum edge cover in graph theory and how does it impact the overall structure of a graph?
A minimum edge cover in graph theory is a set of edges that covers all the vertices in a graph with the fewest number of edges possible. It is significant because it helps identify the smallest number of edges needed to connect all the vertices in a graph. This impacts the overall structure of a graph by showing the essential connections between vertices and highlighting the relationships within the graph.
What is the largest number of vertices in a graph with 35 edges if all vertices are?
36 vertices if all of them are or order two except one at each end.
What is the longest simple path that can be found in a given graph?
The longest simple path in a graph is the path that does not repeat any vertices and has the most number of edges between two distinct vertices.
How can you show that every Hamiltonian cubic graph is 3-edge-colorable?
A cubic graph must have an even number of vertices. Then, a Hamilton cycle (visiting all vertices) must have an even number of vertices and also an even number of edges. Alternatively color this edges red and blue, and the remaining edges green.
What is the largest number of vertices in a graph with 35 edges is all vertices are of degree at least 3?
In a graph, the sum of the degrees of all vertices is equal to twice the number of edges. This is known as the Handshaking Lemma. Therefore, if all vertices in a graph with 35 edges have a degree of at least 3, the sum of the degrees of all vertices must be at least 3 times the number of vertices. Since each edge contributes 2 to the sum of degrees, we have 2 * 35 = 3 * V, where V is the number of vertices. Solving for V, we get V = 70/3 = 23.33. Since the number of vertices must be a whole number, the largest possible number of vertices in this graph is 23.
What is the least possible number of vertices in a graph with 35 edges and all vertices of degree 3?
In a graph where all vertices have a degree of 3, the sum of the degrees of all vertices is equal to twice the number of edges. Therefore, if there are ( n ) vertices, the equation is ( 3n = 2 \times 35 = 70 ). Solving for ( n ) gives ( n = \frac{70}{3} ), which is approximately 23.33. Since ( n ) must be an integer, the least possible number of vertices is 24.
The minimum number of edges in a connected cyclic graph on n vertices is?
n-1
