The complete graph ( K_n ) is Hamiltonian for all ( n \geq 3 ). This means that a Hamiltonian cycle exists in ( K_n ) for any number of vertices ( n ) greater than or equal to 3. For ( n = 1 ) and ( n = 2 ), ( K_n ) does not contain a Hamiltonian cycle, as there aren’t enough vertices to form a closed loop. Thus, ( K_n ) is Hamiltonian for ( n \in {3, 4, 5, \ldots} ).
The automorphism group of a complete bipartite graph K_n,n is (S_n x S_n) semidirect Z_2.
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.
It has a complete lack of any x-intercepts.
can't help about the edge connectivity but a graph is an animal you can see at the zoo - they stand out because they have very long necks and are generally decorated with brown oblongs.
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...
The complete graph ( K_n ) is Hamiltonian for all ( n \geq 3 ). This means that a Hamiltonian cycle exists in ( K_n ) for any number of vertices ( n ) greater than or equal to 3. For ( n = 1 ) and ( n = 2 ), ( K_n ) does not contain a Hamiltonian cycle, as there aren’t enough vertices to form a closed loop. Thus, ( K_n ) is Hamiltonian for ( n \in {3, 4, 5, \ldots} ).
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Yes, finding the longest path in a graph is an NP-complete problem.
The number of triangles in a complete graph with n nodes is n*(n-1)*(n-2) / 6.
The automorphism group of a complete bipartite graph K_n,n is (S_n x S_n) semidirect Z_2.
Yes!
No.
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.
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The complexity of solving the k-color problem on a given graph is NP-complete.
Determining the minimum spanning tree of a graph is not an NP-complete problem. It can be solved in polynomial time using algorithms like Prim's or Kruskal's algorithm.