What is the maximum number of edges in a connected graph with n vertices?
In a directed graph having N vertices, each vertex can connect to N-1 other vertices in the graph(Assuming, no self loop). Hence, the total number of edges can be are N(N-1). There can be as many as n(n-1)/2 edges in the graph if not multi-edge is allowed.
What is the chromatic number of the complete graph with n vertices?
A complete graph with n vertices is n-chromatic, because all its vertices are adjacent. So, χ(Kn) = n and χ(Kn) = 1. Therefore we see that a graph containing a complete graph of r vertices is at least r-chromatic. For example, every graph containing a triangle is at least 3-chromatic.
How many edges are in a connected graph?
The minimum number of edges for undirected connected graph is (n-1) edges. To see this, since the graph is connected then there must be a unique path from every vertex to every other vertex and removing any edge will make the graph disconnected.
What is the minimum possible number of edges in a directed graph?
Adding a directed edge joining the pair of vertices {3, 1} makes the graph strongly connected. Hence, the minimum number of edges required is 1.
How many edges must a connected graph have?
2 Answers. Yes.. The minimum number of edges for undirected connected graph is (n-1) edges. To see this, since the graph is connected then there must be a unique path from every vertex to every other vertex and removing any edge will make the graph disconnected.
How many edges does a connected component have?
If a connected component of a graph has A nodes, then it must contain at least A-1 edges.
What is the minimum number of edges in a connected graph with 16 vertices?
n-1
The minimum number of edges for undirected connected graph is (n-1) edges. To see this, since the graph is connected then there must be a unique path from every vertex to every other vertex and removing any edge will make the graph disconnected.
What is the edge chromatic number of K4?
Toft, Any 7-chromatic graph has K7 or K4,4 as a minor, Combinatorica 25 (2005) 327–353] and Kawarabayashi, Luo, Niu and Zhang [K. Kawarabayashi, R.
What is the minimum number of edges in a connected graph?
That is the resulting graph has less than n vertices in it, so few vertices are left unconnected and thus the graph is not connected which contradicts our assumption. So, it is indeed true that the minimum number of edges in a connected graph with n vertices is n-1.
How many components does a connected graph with n vertices have?
Thus, a graph with n vertices and k edges has at least n − k components. Hence every graph with n vertices and fewer than n − 1 edges has at least two components, and is disconnected. Therefore every connected graph with n vertices must have at least n − 1 edges; the path P n is an example of such a graph. ◼
How do you find a k-connected graph with n vertices?
If k is odd and n is odd, number the vertices 0, 1, …, n − 1, and connect vertex i to vertex i + n − 1 2 for 0 ≤ i ≤ n − 1 2. The result is a k -connected graph on n vertices with the minimum possible number of edges: ⌈ k n 2 ⌉.
How do you increase the number of edges in a graph?
Suppose there were two strongly connected components having m and n vertices where m < n. Now if you remove a vertex from the one having m vertices and add it to the other component, then effectively you have removed m − 1 edges from the first graph and added n edges to the second graph. So, there is a net gain in the number of edges.