Discrete Test

subgraph

: A subgraph of a graph G = (V,E) is a graph (W,F), where W ⊂ V and F ⊂ E. A subgraph H of G is a proper subgraph of G if H ≠ G.

Theorem: A connected multigraph

A connected multigraph with at least two vertices has an Euler circuit if and only if each of its vertices has an even degree and it has an Euler path if and only if it has exactly two vertices of odd degree

Hamilton path,

A simple path in a graph G that passes through every vertex exactly once

loop Adjacency Matrices

Adjacency matrices can also be used to represent graphs with loops and multiple edges. A loop at the vertex vi is represented by a 1 at the (i, j)th position of the matrix. When multiple edges connect the same pair of vertices vi and vj, (or if multiple loops are present at the same vertex), the (i, j)th entry equals the number of edges connecting the pair of vertices. Example: We give the adjacency matrix of the pseudograph shown here using the ordering of vertices a, b, c, d.

Theorem 2:

An undirected graph has an even number of vertices of odd degree.

undirected graphs

Graphs where the end points of an edge are not ordered

Theorem 1 (Handshaking Theorem)

If G = (V,E) is an undirected graph with m edges, then 2𝑚=∑2_(𝑣∈𝑉)〖deg⁡(𝑣)〗 Proof:Each edge contributes twice to the degree count of all vertices. Hence, both the left-hand and right-hand sides of this equation equal twice the number of edges. Think about the graph where vertices represent the people at a party and an edge connects two people who have shaken hands.

Ore's Theorem

If G is a simple graph with n ≥ 3 vertices such that deg(u) + deg(v) ≥ n for every pair of nonadjacent vertices, then G has a Hamilton circuit

Dirac's Theorem

If G is a simple graph with n ≥ 3 vertices such that the degree of every vertex in G is ≥ n/2, then G has a Hamilton circuit.

subgraph induced

Let G = (V, E) be a simple graph. The subgraph induced by a subset W of the vertex set V is the graph (W,F), where the edge set F contains an edge in E if and only if both endpoints are in W.

Handshaking Theorem EX: How many edges are there in a graph with 10 vertices of degree six?

Solution: Because the sum of the degrees of the vertices is 6 10 = 60, the handshaking theorem tells us that 2m = 60. So the number of edges m = 30.

Handshaking Theorem EX If a graph has 5 vertices, can each vertex have degree 3?

Solution: This is not possible by the handshaking thorem, because the sum of the degrees of the vertices 3 5 = 15 is odd.

Adjacency Matrices

Suppose that G = (V, E) is a simple graph where |V| = n. Arbitrarily list the vertices of G as v1, v2, ... , vn. The adjacency matrix AG of G, with respect to the listing of vertices, is the n × n zero-one matrix with 1 as its (i, j)th entry when vi and vj are adjacent, and 0 as its (i, j)th entry when they are not adjacent.

Definition 3 (degree)

The degree of a vertex in a undirected graph is the number of edges incident with it, except that a loop at a vertex contributes two to the degree of that vertex. The degree of the vertex v is denoted by deg(v).

Definition 2 (neighborhood)

The set of all neighbors of a vertex v of G = (V, E), denoted by N(v), is called the neighborhood of v. If A is a subset of V, we denote by N(A) the set of all vertices in G that are adjacent to at least one vertex in A. So,

isomorphic

The simple graphs G1 = (V1, E1) and G2 = (V2, E2) are isomorphic if there is a one-to-one and onto function f from V1 to V2 with the property that a and b are adjacent in G1 if and only if f(a) and f(b) are adjacent in G2 , for all a and b in V1 . Such a function f is called an isomorphism. Two simple graphs that are not isomorphic are called nonisomorphic.

union

The union of two simple graphs G1 = (V1, E1) and G2 = (V2, E2) is the simple graph with vertex set V1 ⋃ V2 and edge set E1 ⋃ E2. The union of G1 and G2 is denoted by G1 ⋃ G2.

Definition 1 (neighbors)

Two vertices u, v in an undirected graph G are called adjacent (or neighbors) in G if there is an edge e between u and v. Such an edge e is called incident with the vertices u and v and e is said to connect u and v.

simple graph G is bipartite if ?

V can be partitioned into two disjoint subsets V1 and V2 such that every edge connects a vertex in V1 and a vertex in V2. In other words, there are no edges which connect two vertices in V1 or in V2.

tree

a connected undirected graph with no simple circuits.

Euler circuit

a graph G is a simple circuit containing every edge of G. An Euler circuit begins with a vertex a and continues with an edge incident with a, say {a, b}. The edge {a, b} contributes one to deg(a). Each time the circuit passes through a vertex it contributes two to the vertex's degree. Finally, the circuit terminates where it started, contributing one to deg(a). Therefore deg(a) must be even. We conclude that the degree of every other vertex must also be even.

A complete bipartite graph Km,n

a graph that has its vertex set partitioned into two subsets V1 of size m and V2 of size n such that there is an edge from every vertex in V1 to every vertex in V2.

forest

a graph that has no simple circuit, but is not connected. Each of the connected components in a forest is a tree.

path

a sequence of edges that begins at a vertex of a graph and travels from vertex to vertex along edges of the graph. As the path travels along its edges, it visits the vertices along this path, that is, the endpoints of these.

Hamilton circuit.

a simple circuit in a graph G that passes through every vertex exactly once

adjacency list

be used to represent a graph with no multiple edges by specifying the vertices that are adjacent to each vertex of the graph.

directed graph / digraph G = (V,E)

consists of a nonempty set V of vertices (or nodes) and a set E of directed edges (or arcs). Each edge is associated with an ordered pair of vertices. The directed edge associated with the ordered pair (u,v) is said to start at u and end at v.

graph G = (V,E)

consists of a nonempty set V of vertices (or nodes) and a set E of edges. Each edge has either one or two vertices associated with it, called its endpoints. An edge is said to connect its endpoints.

A cycle Cn for n ≥ 3

consists of n vertices v1, v2 ,⋯ , vn, and edges {v1, v2}, {v2, v3} ,⋯ , {vn-1, vn}, {vn, v1}.

complete graph on n vertices

denoted by Kn, is the simple graph that contains exactly one edge between each pair of distinct vertices.

The out-degree of v,

denoted deg+(v), is the number of edges with v as their initial vertex. Note that a loop at a vertex contributes 1 to both the in-degree and the out-degree of the vertex.

in-degree of a vertex v

denoted deg−(v), is the number of edges which terminate at v.

influence graphs

directed graphs where there is an edge from one person to another if the first person can influence the second person

simple graph

each edge connects two different vertices and no two edges connect the same pair of vertices.

A directed multigraph

have multiple directed edges. When there are m directed edges from the vertex u to the vertex v, we say that (u,v) is an edge of multiplicity m.

circuit

if it begins and ends at the same vertex (u = v) and has length greater than zero.

simple path

if it does not contain the same edge more than once.

connected

if there is a path between every pair of vertices. An undirected graph that is not connected is called disconnected. We say that we disconnect a graph when we remove vertices or edges, or both, to produce a disconnected subgraph.

Euler path

in G is a simple path containing every edge of G. By the same reasoning, we see that the initial vertex and the final vertex of an Euler path have odd degree, while every other vertex has even degree. So, a graph with an Euler path has exactly two vertices of odd degree.

An n-dimensional hypercube, or n-cube, Qn

is a graph with 2n vertices representing all bit strings of length n, where there is an edge between two vertices that differ in exactly one bit position.

edges

line segment between two nodes

multigraph

may have multiple edges connecting the same two vertices. When m different edges connect the vertices u and v, we say that {u,v} is an edge of multiplicity m.

pseudograph

may include loops, as well as multiple edges connecting the same pair of vertices.

Simple Undirected Graph Multiple Edges allowed? Loops Allowed?

no / no

Simple directed graph Multiple Edges allowed? Loops Allowed?

no / no

simple directed graph

no loops and no multiple edges.

wheel Wn

obtained by adding an additional vertex to a cycle Cn for n ≥ 3 and connecting this new vertex to each of the n vertices in Cn by new edges.

connected component

of a graph G is a connected subgraph of G that is not a proper subgraph of another connected subgraph of G. A graph G that is not connected has two or more connected components that are disjoint and have G as their union.

vertices / nodes

points on a graph

directed graph is weakly connected

there is a path between every two vertices in the underlying undirected graph, which is the undirected graph obtained by ignoring the directions of the edges of the directed graph.

directed graph is strongly connected

there is a path from a to b and a path from b to a whenever a and b are vertices in the graph.

friendship graphs

undirected graphs where two people are connected if they are friends (in the real world, on Facebook, or in a particular virtual world, and so on.)

collaboration graph

undirected graphs where two people are connected if they collaborate in a specific way

Multigraph undirected Multiple Edges allowed? Loops Allowed?

yes / no

Multigraph directed Multiple Edges allowed? Loops Allowed?

yes / yes

pseudograph undirected Multiple Edges allowed? Loops Allowed?

yes / yes

mixed graph directed and undirected Multiple Edges allowed? Loops Allowed?

yes /yes