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graph
a mathematical model that consists of arcs/edges and nodes/vertices
network
a weighted graph i.e. a graph with numbers associated to its edges
vertex set
list of all the vertices in a given graph
edge set
a list of all the edges in a given graph
subgraph
A part of the original graph (vertices and edges also belong to original graph)
degree/valency/order of a vertex
number of edges incident to it
pendant
a node of degree one
isolated
a node of degree zero
walk
a route through a graph along edges from one vertex to the next
path
a walk with no repeated vertices
trail
a walk with no repeated edges
cycle
a walk in which the end vertex is the same as the start vertex and no other vertex is visited more than once (or a trail that starts and ends at the same vertex)
Hamiltonian cycle
A cycle that visits every vertex of a graph
two vertices are connected if...
...there is a path between them
a graph is connected if...
...all its vertices are connected
connected components
subgraphs that are connected
loop
an edge that starts and finishes at the same vertex
how does a loop affect the degree of a vertex
the vertex the loop originates from has a degree of 2 and not 1 due to it. this is because the loop is incident at the vertex twice.
simple graph
a graph with no loops or multiple edges between vertices
directed graph (digraph)
a graph with a direction associated with the edges
Euler's handshaking lemma
in any undirected graph, the sum of the degrees of the vertices is equal to 2x the number of edges. As a consequence, the sum of the degrees must always be even.
how is euler's handshaking lemma used?
to determine whether a graph can exist - if the sum of the degrees is odd, that implies the number of edges is not an integer which is impossible
in any undirected graph there must be...
an even amount of vertices with an odd degree
tree
a simple connected graph with no loops or circuits
spanning tree
a tree which includes all the vertices of the graph
complete graph
a graph where every vertex is directly connected by a single edge to each and every other vertex
how is a complete graph denoted?
Kₙ where n is the number of vertices in the graph
isomorphic graphs
graphs that show the same information but are drawn differently
what makes two graphs isomorphic?
- they have the same number of vertices with the same degree
- the vertices are connected in the same way
- the vertices may have different labels as they are not the same graph but still an equivalence
adjacency matrix
a matrix that provides information about the connections between the vertices in a graph
what should each entry be for the adjacency matrix of an unweighted graph?
the number of arcs joining two points
describe how a loop from point A to A is depicted in an adjacency matrix for an undirected graph
it counts as two arcs as they can go in either direction ∴ the entry would be 2
(it is a similar principle to how e.g. AB could be 1 ∴ BA would also be 1)
distance matrix
the matrix associated with a weighted graph
what should each entry be for a distance matrix?
the weight of the arc joining two nodes
what if there are multiple arcs joining two vertices in a weighted graph? what should the entry be in the distance matrix + why?
the smallest weight; this is because when an algorithm is applied to the matrix to e.g. find the shortest route, the least value is desired.
minimum spanning tree (MST)
a spanning tree such that the total length of its edges is as small as possible
what is another name for a minimum spanning tree?
minimum connector
Eulerian circuit
A trail that starts and ends at the same vertex and traverse every arc no more than once
Semi-eulerian circuit
A trail that traverses every arc no more than once, but doesn’t start and end at the same vertex
Traits of a Eulerian graph
all vertices are even
connected
contains a Eulerian circuit
Traits of a semi-Eulerian graph
exactly two vertices are odd
connected
contains a semi-Eulerian circuit
Note 
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