Degree – No. of edges directly connected to a node
Isolated Vertex – No connected edges, 0 Degree
Loop – Node with a loop has a 2 Degree
Isomorphic – Networks that are equivalent (same no. vertices & edges)
In matrix representation, the following rules apply:
- The sum of a row (or a column) gives a degree of that vertex, except where a loop is present. to the
sum of the row or column.
- If an entire row and its corresponding entire column have only 0’ s then that vertex is isolated.
Type of Graph
Degenerate/Null
Definition & Example
No Edges
Complete
All vertices connected to each other (no parallel
edges or loops)
Planar
No edges that cross
Simple Graph
No more than edge between any 2 vertices (graph
without loops/multiple edges)
Eulers Formula (Applied to Verify if Graph is
Planar)
Walk
Connected Graphs
Bridge
Trail
Euler Trail
- Exists if 2 vertices have odd degrees
Circuit
Euler Circuit
- Doesn’t exist in Planar graphs which a
vertex has odd degree
Path
Cycle
Any route taken through a network
A walk between all vertices is possible
An edge in a connected graph which if removed
will no longer be connected
No Edges are repeated
Can pass through a vertex more than once
Uses every edge exactly once
A trail (no edges repeating) beginning & ending at
the same vertex
A Euler trail (passing through every edge once)
where starting & ending vertices are the same.
A walk where no vertices are repeated
A path (no vertices repeated) begin & end at the
same vertex
Hamiltonian Path
Hamiltonian Cycle
Every vertex used exactly once (not all edges need
to be used)
A Hamiltonian Path (every vertex used once)
starting & finishing at the same vertex
Trees can’t contain:
- Loops
- Parallel/Multiple Edges
- Cycles
Spanning Tree
- A tree that’s a subgraph that includes all the vertices of the original path
Minimum Spanning Tree
- Covers the shortest distance for a network (consider smallest weighted edges)
Maximum Spanning Tree:
- Covers the greatest distance for a network (consider largest weighted edges)
Dijkstra’s Algorithm:
- Method Determining shortest path between vertices
EST (Earliest Starting Time):
- Earliest time an activity can be started after all prior activities have been completed
- Written in the Triangle
Forward Scanning:
- Calculating EST for each activity & earliest completion time for entire project
E.g.
Critical Path:
- The path through the network which cannot be delayed
Float Time:
- Difference in time between paths that cannot be delayed by those which can
LST (Latest Start Time):
- The latest time an activity can begin (be delayed until)
E.g.
Backward Scanning:
- Used to calculate the critical path in a network
- Begin at the last node and move backwards subtracting each edge from EST
Critical Path Analysis Rules:
- 2 vertices can be connected by a max. of 1 edge
- An activity must be represented by exactly 1 edge
Dummy Activity:
- An edge that must be added to a avoid with 2+ activities have the same name or occurring in
parallel
- Rule 1: 2 vertices can be connected directly by 1 max edge
- Rule 2: An activity must be represented by 1 edge only
Crashing:
-
Method to speed up completion time of project by shortening critical path
Source – Network Starting Node, Sink – Network ending Node
Flow Capacity – the amount of flow an edge can allow through
Inflow: - total of flow into the network
Outflow – minimum value obtained when comparing inflow to sum of capacities of all edges leaving vertex
Excess Flow Capacity – surplus of capacity of an edge – flow into the edge
The minimum cut is the cut with the minimum value
The maximum flow through a network is equal to the value of the minimum cut.
Bipartite Graph – Nodes can be separate into 2 types; supply & demand
Optimal Allocation – Most Efficient Result
Hungarian Algorithm:
Activity
Earliest Start Time
Duration (Days)
A
B
C
D
E
F
G
H
I
J
K
L
0
0
0
6
5
4
4
13
9
6
5
4
7
4
5
6
2
5
Immediate
Predecessor
A
B
C
C
D
E