Lecture 01-13-17

Order Notation

• 
• can use calculus

Sum of 1 to n

• proof that

Explanation for Roger

• The terms are from 1 to n so half of the terms are >= n / 2 (that’s the value of each term)
• If half the terms are like ^, and there are n terms, there are n / 2 numbers like ^ (that’s how many of those terms there are)
• So, n/2 is the value * n/2 terms with those values = lower bound of n²/4

Graphs

• Representing relationships between objects
• Terminology:
  • edges & vertices
  • nodes & arcs
  • points & lines
• Controversies
  • Smallest graph has 0 or 1 vertex?
  • Empty graph
• Assumption by Spinrad: graphs are undirected
• Special cases:
  • Edge from node to same node
    • reflexive relationships
  • Multiple edges from one node to another
    • assumption in 250: no multiple edges
  • In weighted graphs, length of edge is cost
• Definitions
  • x is a neighbor of y: if you can get from x to y
  • degree of a vertex: number of edges coming out of the vertex
  • n: # vertices
  • m: # edges
  • sum of all vertices (degree(v)) = 2m ??
  • loop = degree + 2

Homework

• To-do:
  • Derive n(n + 1) / 2
  • Review notes and figure out ??
• Completed: