Can a graph have the same number of vertices and edges?
A graph can exist in different forms having the same number of vertices, edges, and also the same edge connectivity. Such graphs are called isomorphic graphs.
How many edges will have for a connected graph with n vertices?
Hence, as you correctly calculate, there are (n2)=(n−1)(n−2)2 edges. Adding any possible edge must connect the graph, so the minimum number of edges needed to guarantee connectivity for an n vertex graph is (n−1)(n−2)2+1.
Which has the same number of vertices and edges?
A cube and a trapezium prism have the same number of faces, vertices and edges.
How do you know if two graphs are equivalent?
Two graphs are equivalent if they have the same set of edges (ex. (A,B),(A,C)). It should be: Two graphs are equal if they have the same vertex set and the same set of edges.
Does there exist a simple graph with 5 vertices?
Does there exist a simple graph with five vertices of the following degrees? No, as the sum of the degrees of an undirected graph is even.
Can a simple graph exist with 15 vertices each of degree five?
Therefore by Handshaking Theorem a simple graph with 15 vertices each of degree five cannot exist.
How many edges does a simple 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 can a simple graph G on n have if it is not connected?
Alternate solution There are exactly k(n−k) edges between vertices in the two pieces. If you add them to your graph, you get a simple graph, which by handshaking lemma, has at most n(n−1)2 edges.
How many (N2) graphs are there without loops or multiple edges?
There are (n 2) = 1 2n(n − 1) pairs of distinct points. If you do not allow loops or multiple edges, each of these pairs determines one possible edge, and you can have any subset of those possible edges. A set with (n 2) members has 2 (n 2) subsets, so there are 2 (n 2) possible graphs without loops or multiple edges.
When a planar graph is drawn without edges crossing?
When a planar graph is drawn without edges crossing, the edges and vertices of the graph divide the plane into regions. We will call each region a face.
How do you prove that a graph has a vertex set?
Proof: First, for any labeled graph, define C(k) to be the component of vertex k in that graph. Now let Wn, S ⊂ Wn denote the subset of connected graphs on n labeled vertices, such that when vertex n is removed, the C(1) has vertex set S.
How many subsets of edges does a kn graph have?
For Kn, there will be n vertices and (n (n-1))/2 edges. To determine how many subsets of edges a Kn graph will produce, consider the powerset as Brian M. Scott stated in a previous comment. If S is a finite set with n elements, then the powerset of S will have 2^n elements where n is the number of elements in the set S.