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