What is the order of graph G?

The order of a graph G is the cardinality of its vertex set, and the size of a graph is the cardinality of its edge set. Given two vertices u and v, if uv ∈ E, then u and v are said to be adjacent.

What is K ‘( G in graph theory?

The vertex connectivity κ(G) (where G is not a complete graph) is the size of a minimal vertex cut. A graph is called k-vertex-connected or k-connected if its vertex connectivity is k or greater. A vertex cut for two vertices u and v is a set of vertices whose removal from the graph disconnects u and v.

How do you prove the connectivity of a graph?

Given a graph with n vertices, prove that if the degree of each vertex is at least (n−1)/2 then the graph is connected. The distance between two vertices in a graph is the length of the shortest path between them. The diameter of a graph is the distance between the two vertices that are farthest apart.

Are K regular graphs K connected?

A regular graph of degree k is connected if and only if the eigenvalue k has multiplicity one.

Is K1 connected?

According to Bogdán Zaválniji’s definition of connectivity, if we take any pair of vertices of a graph and there is path connecting them then the graph is connected. So, if we take K1, the only pair of vertices we can take is the single vertex v. But there is no path connecting v and v. So, how K1 is connected.

How do you show a graph is 2 connected?

A graph is connected if for any two vertices x, y ∈ V (G), there is a path whose endpoints are x and y. A connected graph G is called 2-connected, if for every vertex x ∈ V (G), G − x is connected.

How do you show that a graph is complete?

In the graph, a vertex should have edges with all other vertices, then it called a complete graph. In other words, if a vertex is connected to all other vertices in a graph, then it is called a complete graph.

Is there a k-regular graph with $n-1$ vertices?

True , for k or n even. False , for k and n odd . But we can find a graph with $n-1$ vertices with degree k and one vertex with degree $k-1$. There doesn’t exists a k-regular graph for k and n odd because $k=\\deg(G) = 2*|E(G)| / |V(G)|$ $|E(G)| = k*n/2$, and $|E(G)|= m$ is not a natural number if $n$ and $k$ is odd.

What is the degree of each vertex of a graph?

Every graph has certain properties that can be used to describe it. An important property of graphs that is used frequently in graph theory is the degree of each vertex. The degree of a vertex in G is the number of vertices adjacent to it, or, equivalently, the number of edges incident on it.

How do you determine if a graph is undirected?

An undirected graph is described by a pair ( V, E ) where V is set of vertices and E, the set of edges, is a set of two-element subsets of V. If G = ( V, E) is an undirected graph, define G = ( V, E ). That is,

How do you find the largest clique in a graph?

If the degree of v is n − 1, stop; G is a clique, so the largest clique in G has size n . Otherwise, remove v and all of its edges from G . Find the largest clique in the smaller graph. Report that as the largest clique in G . Let’s run this algorithm on G2 . Iteration 1. Vertex 7 has degree 2. Remove it. Iteration 2.