Discrete Math: Graphs (Basics)

Subgraph

- Where every vertex in H is also in G - Every edge in H is also in G - Every edge has the same endpoints in G that it does in H

Formally a graph is defined as?

A Graph G consists of TWO finite sets: - a nonempty set V(G). Which is each vertex: ex: {v₁, v₂, v₃} - a nonempty set E(G). Which is each edge. ex: {e₁, e₂, e₃} -Each element of E(G) corresponds to either 1 or 2 vertices

Simple Graph

A graph containing no parallel or loop edges. All edges therefore, have 2 endpoints because they each connect two vertices.

Complete Graph

A graph in which all vertex pairs are connected by edges. Denoted: K sub n where n is the number of vertices

Directed Graph (digraph)

A graph where each edge is associate with an ordered pair of vertices. ex: e₁ is associated with {A, B}, e₂ with {B, C} and e₃ with {C, A}

What is a graph

A series of connected dots called vertices (nodes). The lines connecting the vertices are called edges.

Isolated Vertex

A vertex with no edges connected to it. No edges are incident on it in other words.

Complete Bipartite Graph

Basically all vertices pairs of different sets are connected with edges. No vertices from the same set can be joined.

Degree

Denoted deg(v) For a single vertex, the degree is the number of edges incident on it. A loop is incident twice and therefore has degree 2. An isolated vertex is degree 0.

Total Degree of G

The sum of the degree on every vertex in the graph.

The Handshake Theorem? What corollary can be extrapolated from this?

The total number of degrees G = 2x where x = the total number of edges in G The total number of degrees is therefore even

What does it mean for an edge to be incident on?

This just states which vertices it is connected to. e₁ is incident on {v₁ and v₂}

Parallel Edges

When two edges connect the same pair of vertices. Or rather when two vertices share the same endpoint.

When an edge only has one endpoint this is known as ____

a loop the edge only connects the vertex back to itself. It is said to be adjacent to itself.

Two vertices connected by the same edge are called _____

adjacent

The correspondence between the edges and the vertices is known as?

the edge-endpoint function.