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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.


Is the complete graph on 5 vertices planar?

No, the complete graph of 5 vertices is non planar. because we cant make any such complete graph which draw without cross over the edges . if there exist any crossing with respect to edges then the graph is non planar.Note:- a graph which contain minimum one edge from one vertex to another is called as complete graph...


How many Hamiltonian circuits not counting reversals are there in a complete graph with 7 vertices?

In a complete graph with ( n ) vertices, the number of distinct Hamiltonian circuits, not counting reversals, is given by ( \frac{(n-1)!}{2} ). For a complete graph with 7 vertices, this calculation is ( \frac{(7-1)!}{2} = \frac{6!}{2} = \frac{720}{2} = 360 ). Therefore, there are 360 distinct Hamiltonian circuits in a complete graph with 7 vertices when not considering reversals.


What makes a complete graph?

A complete graph is a type of graph in which every pair of distinct vertices is connected by a unique edge. In a complete graph with ( n ) vertices, denoted as ( K_n ), there are exactly ( \frac{n(n-1)}{2} ) edges. This means that every vertex is adjacent to every other vertex, resulting in a highly interconnected structure. Complete graphs are often used in graph theory to illustrate maximum connectivity among a set of points.


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.


How many subgraphs with at least one vertex does a complete graph of 3 vertices k3 have?

one vertex: 3 two vertices: 6 three vertices: 8 total 17


Show that the star graph is the only bipartiate graph which is a tree?

A star graph, call it S_k is a complete bipartite graph with one vertex in the center and k vertices around the leaves. To be a tree a graph on n vertices must be connected and have n-1 edges. We could also say it is connected and has no cycles. Now a star graph, say S_4 has 3 edges and 4 vertices and is clearly connected. It is a tree. This would be true for any S_k since they all have k vertices and k-1 edges. And Now think of K_1,k as a complete bipartite graph. We have one internal vertex and k vertices around the leaves. This gives us k+1 vertices and k edges total so it is a tree. So one way is clear. Now we would need to show that any bipartite graph other than S_1,k cannot be a tree. If we look at K_2,k which is a bipartite graph with 2 vertices on one side and k on the other,can this be a tree?


Difference between a directed graph and an undirected graph in a computer program?

In an undirected graph, an edge is an unordered pair of vertices. In a directed graph, an edge is an ordered pair of vertices. The ordering of the vertices implies a direction to the edge, that is that it is traversable in one direction only.


What is subgraph in given graph?

If all the vertices and edges of a graph A are in graph B then graph A is a sub graph of B.


How can i use a graph to find the number of vertices in a octagonal pyramid?

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.


How many spanning trees can be drawn with 5 labeled vertices?

No of spanning trees in a complete graph Kn is given by n^(n-2) so for 5 labelled vertices no of spanning trees 125


What is the maximum number of distinct edges in an undirected graph with N vertices?

Let G be a complete graph with n vertices. Consider the case where n=2. With only 2 vertices it is clear that there will only be one edge. Now add one more vertex to get n = 3. We must now add edges between the two old vertices and the new one for a total of 3 vertices. We see that adding a vertex to a graph with n vertices gives us n more edges. We get the following sequence Edges on a graph with n vertices: 0+1+2+3+4+5+...+n-1. Adding this to itself and dividing by two yields the following formula for the number of edges on a complete graph with n vertices: n(n-1)/2.