graph theory terminology

what is the total number of edges in a simple complete graph

(v(v-1))/2

properties of tree graphs

1. it has its vertices connected by exactly one path 2. if an edge is removed the graph is disconnected 3. a tree with order n has n-1 edges

Eulerian Circuit and conditions

A circuit (a walk that begins and ends at the same vertex, and has no repeated edges) that contains every edge of a graph. a closed eularian trail. A graph must only have vertices of even degrees

tree

A connected graph with no cycles (a walk that begins and ends at the same vertex, and has no other repeated vertices)

State Kuratowski's theorem

A graph is not planar iff it contains a homeomorphic to K3,3 or K5

planar graph

A graph that can be drawn in the plane without any edges crossing

disconnected graph

A graph that has at least one pair of vertices not joined by a path

bipartite graph

A graph whose vertices can be divided into two sets and in which edges always join a vertex from one set to a vertex from the other set

complement of a graph G

A graph with the same vertices as G but which has an edge between any two vertices if and only if G does not

subgraph

A graph within a graph

walk

A sequence of linked edges (includes both trails and paths) with finite steps

complete graph

A simple graph where every vertex is connected to every other vertex by a single edge

Eulerian Trail and conditions

A trail (a walk in which no edge appears more than once) that contains every edge of a graph. a graph must have exactly two vertices of an odd degree.

weighted tree

A tree in which each edge is allocated a number or weight

Chinese postman problem

Eulerian circuit with min weight; vertices with odd degree must be walked through twice; then sole by inspection to determine the shortest route (walk) around a weighted graph using each edge (at least once, returning to the starting vertex)

State the Handshaking Lemma

Every finite undirected graph has an even number of vertices with an odd degree. This is because the sum of the degrees of the vertices is twice the size of the graph.

Isomorphic graphs

Graphs that show the same information but are drawn differently

Degree of a vertex

The number of edges joined to the vertex; a loop contributes two edges, one for each of its end points.

When is a connected graph Eularian?

When it has a eularian circuit. it is semi-eularian if it contains a eularian trail

Complete bipartite graph K(m,n)

a bipartite graph in which every vertex in one set is joined to every vertex in the other set.

Hamiltonian cycle

a cycle (a walk that begins and ends at the same vertex, and has no other repeated vertices) that contains all vertices of the graph. if the graph is complete it contains (n-1)! hamiltonian cycles

weighted graph

a graph in which each edge is allocated a number or a weight

When is a graph bipartite?

a graph is bipartite iff each circuit of the graph has an even length (and the vertices can be divided into two groups depending on edges and degrees)

connected graph

a graph that has a path (a walk with no repeated vertices) joining every pair of vertices

Simple graph and property

a graph with no loops or multiple edges (has a min. degree of 1 and a max. degree of n-1). therefore with n vertices, two vertices must have the same degree

Graph isomorphism between two simple graphs G and H

a one-to-one correspondence between vertices of G and H such that a pair of vertices in G is adjacent if and only if the corresponding pair in H is adjacent

Hamiltonian Path

a path (a walk with no repeated vertices) that contains all vertices of the path

identity arising from the pigeonhole principle

a simple directed graph always has two vertices of the same degree. This is because a simple connected graph will have a min. degree of 1 and a max of n-1, hence with n vertices, two vertices must have the same degree.

minimum spanning tree

a spanning tree of a weighted graph that has the minimum total weight

Spanning tree of a graph and property

a subgraph containing every vertex of the graph, which is also a tree (i.e. a connected graph with no cycles). must have two degrees of 1

trail

a walk in which no edge appears more than once, vertices can be used repeatedly though

cycle

a walk that begins and ends at the same vertex, and has no other repeated vertices

circuit

a walk that begins and ends at the same vertex, and has no repeated edges

path

a walk with no repeated vertices

when does a graph have a eularian circuit

all vertices must be of an even degree

loop

an edge whose endpoints are joined to the same vertex

closed trail

circuit

graph

consists of a set of vertices and a set of edges; an edge joins the vertices

closed path

cycle

travelling salesman problem

least weight Hamiltonian cycle upper bound is twice the minimum spanning tree lower bound is found by deleting a vertex, then finding a min. spanning tree for the remaining graph and then adding the shortest ages (lower bound is not necessarily a solution). graph must be complete for a solution to exist

when is a graph connected in relation to number of edges and vertices

n-1 ≤ m ≤ 0.5(n)(n-1) with m vertices and n edges. consequently, the graph with m vertices is connected if it has more than 0.5(n-1)(n-2) edges.

size

number of edge

order

number of vertices

meaning of Hamiltonian in general

the path/cycle contains all vertices of the graph

state the identity related to sum of degrees of vertices

the sum of the degrees of the vertices of a graph is equal to twice the number of edges. Consequently, the sum of the degrees of the vertices is always an even number.

what is purpose of Kruskal's algorithm (and two other graph algorithms)

to find a spanning tree for a graph (there is also depth first search, and breadth first search)

what's the purpose of dijkstra's algorithm

to find the shortest (lightest) path between two vertices of a weighted graph.

Homeomorphic Graphs

two graphs are homeomorphic if they can be obtained from the same graph by subdivision of edges

multiple edges

two or more edges connecting the same two vertices

when do a hamiltonian graph exist?

when a graph has at least 0.5(n-1)(n-2) +2 edges and the degree of each vertex is greater than 0.5n where n is the order of the graph

when does a eularian trail exist

when it has exactly two vertices of an odd degree