What is K in graph theory?

What is K in graph theory?

In graph theory, a connected graph G is said to be k-vertex-connected (or k-connected) if it has more than k vertices and remains connected whenever fewer than k vertices are removed.

What is handshaking theorem in graph theory?

Handshaking theorem states that the sum of degrees of the vertices of a graph is twice the number of edges. Since the degree of a vertex is the number of edges incident with that vertex, the sum of degree counts the total number of times an edge is incident with a vertex.

What is bipartite graph example?

A graph G=(V, E) is called a bipartite graph if its vertices V can be partitioned into two subsets V1 and V2 such that each edge of G connects a vertex of V1 to a vertex V2. It is denoted by Kmn, where m and n are the numbers of vertices in V1 and V2 respectively. Example: Draw the bipartite graphs K2, 4and K3 ,4.

What is a 3 regular graph?

A 3-regular graph is known as a cubic graph. A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number l of neighbors in common, and every non-adjacent pair of vertices has the same number n of neighbors in common.

How do you prove a graph is k-connected?

(Expansion Lemma) If G is a k-connected graph, and G’ is obtained from G by adding a new vertex y with at least k neighbors in G, then G’ is k-connected. Proof: Let S be a vertex set that: (a) Is a vertex cut for G’; or (b) has n(G’–S)=1. If (b) is true, then |S∩V(G)| ≥ k; therefore |S| ≥ k+1.

What is a K2 3 graph?

Bipartite Complete Graph: A graph is a bipartite complete graph if its vertices can be partitioned into two disjoint nonempty sets V1 and V2 such that two vertices x and y are adjacent if and only if x ∈ V1 and y ∈ V2. If |V1| = m and |V2| = n, such a graph is denoted Km,n. Therefore, the graph in Figure 2 is K2,3.

What is the difference between Euler path and Euler circuit?

An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. ▶ An Euler path starts and ends at different vertices.

How do you prove handshaking theorem?

Let’s verify the Handshaking theorem. A directed graph is a graph G = (V,E) for which each edge represents an ordered pair of vertices. If e = (u, v) is an edge of a directed graph, then u is called the start vertex of the edge, while v is called the end vertex of the edge.

What is a K regular bipartite graph?

A graph is said to be bipartite if its set of vertices can be partitioned into two. subsets SO that each edge of the graph has its ends in different subsets. A graph is. said to be regular, or more precisely k-regular where k is a positive integer, if. each vertex is a degree of k.

What is difference between bipartite graph and complete bipartite graph?

By definition, a bipartite graph cannot have any self-loops. For a simple bipartite graph, when every vertex in A is joined to every vertex in B, and vice versa, the graph is called a complete bipartite graph. If there are m vertices in A and n vertices in B, the graph is named Km,n. Fig.

What is the size of K regular graph?

We study the k-diameter of k-regular k-connected graphs. Among other results, we show that every k-regular k-connected graph on n vertices has k-diameter at most n/2 and this upper bound cannot be improved when n = 4k – 6 + i(2k -4).

What is a K5 graph?

K5 is a nonplanar graph with the smallest number of vertices, and K3,3 is the nonplanar graph with smallest number of edges. Thus both are the simplest nonplanar graphs.