{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture19

# lecture19 - COMPSCI 102 Discrete Mathematics for Computer...

This preview shows pages 1–14. Sign up to view the full content.

COMPSCI 102 Discrete Mathematics for Computer Science

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Graphs II Lecture 18
Recap

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Theorem: Let G be a graph with n nodes and e edges The following are equivalent: 1. G is a tree (connected, acyclic) 3. G is connected and n = e + 1 4. G is acyclic and n = e + 1 5. G is acyclic and if any two non-adjacent points are joined by an edge, the resulting graph has exactly one cycle 2. Every two nodes of G are joined by a unique path
The number of labeled trees on n nodes is n n-2 Cayley’s Formula

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
A graph is planar if it can be drawn in the plane without crossing edges
Euler’s Formula If G is a connected planar graph with n vertices, e edges and f faces, then n – e + f = 2

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
A coloring of a graph is an assignment of a color to each vertex such that no neighboring vertices have the same color Graph Coloring
Spanning Trees A spanning tree of a graph G is a tree that touches every node of G and uses only edges from G Every connected graph has a spanning tree

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Finding Optimal Trees Trees have many nice properties (uniqueness of paths, no cycles, etc.) We may want to compute the “best” tree approximation to a graph If all we care about is communication , then a tree may be enough. We want a tree with smallest communication link costs
Finding Optimal Trees Problem: Find a minimum spanning tree , that is, a tree that has a node for every node in the graph, such that the sum of the edge weights is minimum

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
4 8 7 9 6 11 9 5 8 7 Tree Approximations
Finding an MST: Kruskal’s Algorithm Create a forest where each node is a separate tree Make a sorted list of edges S

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 34

lecture19 - COMPSCI 102 Discrete Mathematics for Computer...

This preview shows document pages 1 - 14. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online