CS 291 Graph Theory

What are parallel edges?

Parallel edges are when you have distinct edges associated with the same nodes. Ex. e1 and e2 are associated with {v1 and v2}.

What does it mean for "v" and "w" to be adjacent vertices?

Vertices are adjacent vertices when they are connected by an edge.

What is an Euler Cycle? How to know if an Euler Cycle exists?

A Euler cycle is a cycle that visits all edges once, and all vertices as many times as you want. If G is a connected graph and every vertex has an even degree, then G has a Euler cycle.

Define connected graph.

A graph in which we can get from any vertex to any other vertex on a path.

What is a loop?

A loop is an edge incident on a single node.

What is a simple path?

A path from v to w with no repeated vertices.

What is a cycle?

A path of nonzero length from v to v with no repeated edges.

What is a simple graph

A simple graph is a graph that doesn't have loops or parallel edges.

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

A vertex is an incident on an edge when said vertex is one of the vertices that the edge is an instant on. For example, vertex v1 is an incident on e1 if e1 = (v1,v2).

What is a weighted graph.

A weighted graph is when edges are given positive values that reference some sort information, such as time, prices, and etc. For example, e ∈ E and e is labeled k, so we say the edge e has a weight of k.

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

An edge is an incident on a vertex when two edges share one vertex. For example, e1 and e2 are incidents on v2 if e1 = (v1,v2) and e2 = (v2,v3) since they share v2.

What is an isolated vertex?

An isolated vertex is a vertex that is not an incident on any edge.

Define subgraph

Let G = (V, E) be a graph. We call (V′, E′) a subgraph of G if... (a) V′ ⊆VandE′ ⊆E. (b) For every edge e′ ∈ E′, if e′ is incident on v′ and w′, then v′, w′ ∈ V′.

Define n-cude.

N-cube is a model for parallel computation. Every edge connects two nodes if the binary representation of their labels differ in exactly one bit. This creates a wireframe cube shape.

What is the complete bipartite graph on m and n vertices? How is it denoted?

The complete bipartite graph on "m" and "n" vertices is the simple graph whose vertex set, V( ) , is partitioned into sets V1( ) with "m" vertices and V2( ) with "n" vertices in which the graph's set of edges, E( ), consists of all edges of the form (v1 , v2 ) with v1∈ V1 and v2∈V2. The complete bipartite graph is denoted by Km,n.

What is the complete graph on "n" vertices? How is it denoted?

The complete graph on "n" vertices is the simple graph with "n" nodes and for every distinct node there exists an edge. The complete graph is denoted as Kn.

Define bipartite graph.

A bipartite graph is a graph where within the graph's set of vertices V( ) there exists two subsets of vertices V1( ) and V2( ). The intersect of these two must be an empty set, the union of these two subsets must equal V( ) ,and each e ∈ E the incident vertices for these edges must have one vertex from V1( ) and the other from V2( ).

What is a simple cycle?

A cycle from v to v in which, except for the beginning and ending vertices that are both equal to v, there are no repeated vertices.

Define directed graph.

A directed graph is defined as having two sets, V( ) which contains vertices, and E ( ) which contains edges. For each e ∈ E, they must be associated with an ordered pair of vertices that come from V( ). If there is a unique edge e with vertices x and y, we write e = (x,y). This notation denotes an edge or "incident" between these vertices.

Define undirected graph.

An undirected graph is defined as having two sets, V( ) which contains vertices, and E ( ) which contains edges. For each e ∈ E, they must be associated with an unordered pair of vertices they come from V( ). If there is a unique edge e with vertices x and y, we write e = (x,y) or e = (y,x). This notation denotes an edge or "incident" between these vertices.

Define length of path in a weighted graph

The length of a path in a weighted graph is the sum of the weight of each edge in the path.

What is a Hamiltonian cycle? How to tell if a Hamiltonian cycle exists?

a cycle in a graph G that contains each vertex in G exactly once, except for the starting and ending vertex that appears twice, a Hamiltonian cycle. Not straight forward like Euler cycles. However, it needs to be... 1. A connected graph. 2. No vertex can have a degree of less than 2.