COMPSCI 230 Discrete Mathematics for Computer Science Graphs Whats a tree? A tree is a connected graph with no cycles

Tree Not aTree Not a Tree Tree How Many n-Node Trees? 1: 2:

3: 4: 5: Notation In this lecture: n will denote the number of nodes in a graph e will denote the number of edges in a graph Theorem: Let G be a graph with n nodes and e edges

The following are equivalent: 1. G is a tree (connected, acyclic) 2. Every two nodes of G are joined by a unique path 3. G is connected and n = e + 1 4. G is acyclic and n = e + 1 5. G is acyclic and if any two nonadjacent points are joined by adding a new edge, the resulting graph has exactly one cycle To prove this, it suffices to show

123451 1 2 1. G is a tree (connected, acyclic) 2. Every two nodes of G are joined by a unique path Proof: (by contradiction) Assume G is a tree that has two nodes connected by two different paths: Then there exists a cycle!

23 2. Every two nodes of G are joined by a unique path 3. G is connected and n = e + 1 Proof: (by induction) Assume true for every graph with < n nodes

Let G have n nodes and let x and y be adjacent G1 x y G2 Let n1,e1 be number of nodes and edges in G1

Then n = n1 + n2 = e1 + e2 + 2 = e + 1 3 43. G is connected and n = e + 1 4. G is acyclic and n = e + 1 Proof: (by contradiction) Assume G is connected with n = e + 1, and G has a cycle containing k nodes k nodes Note that the cycle has k nodes and k edges Starting from cycle, add other nodes and

edges until you cover the whole graph Number of edges in the graph will be at least n Corollary: Every nontrivial tree has at least two endpoints (points of degree 1) Proof: Assume all but one of the points in the tree have degree at least 2 In any graph, sum of the degrees =2e Then the total number of edges in the

tree is at least (2n-1)/2 = n - 1/2 > n - 1 How many labeled trees are there with three nodes? 1 2 3 1

3 2 2 1 3

How many labeled trees are there with four nodes? a c b d How many labeled trees are

there with five nodes? 5 labelings 3 4 5 2 2

labelings labelings 4 5 3 125 labeled trees

How many labeled trees are there with n nodes? 3 labeled trees with 3 nodes 16 labeled trees with 4 nodes 125 labeled trees with 5 nodes nn-2 labeled trees with n nodes Cayleys Formula The number of labeled trees on n nodes is nn-2

The proof will use the correspondence principle Each labeled tree on n nodes corresponds to A sequence in {1,2,,n}n-2 (that is, n-2 numbers, each in the range [1..n]) How to make a sequence from a tree?through i from 1 to n-2 Loop Let L be the degree-1 node

with the lowest label Define the ith element of the sequence as the label of the node adjacent to L Delete the node L from the tree Example: 5 1 8 4

3 2 6 1 3 3 4 4 4 7 How to reconstruct the unique tree

from a sequence S: Let I = {1, 2, 3, , n} Loop until S is empty Let i = smallest # in I but not in S Let s = first label in sequence S Add edge {i, s} to the tree Delete i from I Delete s from S Add edge {a,b}, where I = {a,b} S:

1 3 3 4 4 4 I: 1 2 3 4 5 6 7 8 5 1 8

4 3 2 6 7 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 A graph is planar if it can be drawn in the plane without crossing edges Examples of Planar Graphs

= Faces A planar graph splits the plane into disjoint faces 4 faces Eulers Formula If G is a connected planar graph

with n vertices, e edges and f faces, then n e + f = 2 Rather than using induction, well use the important notion of the dual graph Dual = put a node in every face, and an edge between every adjacent face Let G* be the dual

graph of G Let T be a spanning tree of G Let T* be the graph where there is an edge in dual graph for each edge in G T Then T* is a spanning tree for G* n = eT + 1 n + f = eT + eT* + 2 =e+2

f = eT* + 1 Corollary: Let G be a simple planar graph with n > 2 vertices. Then: 1. G has a vertex of degree at most 5 2. G has at most 3n 6 edges Proof of 1 (by contradiction): In any graph, (sum of degrees) = 2e Assume all vertices have degree 6 Then e 3n Furthermore, since G is simple, 3f 2e

So 3n + 3f 3e, 3n + 3f = 3e + 6, and 3e + 6 3e Graph Coloring 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 Arises surprisingly often in CS Register allocation: assign temporary

variables to registers for scheduling instructions. Variables that interfere, or are simultaneously active, cannot be assigned to the same register Theorem: Every planar graph can be 6colored Proof Sketch (by induction): Assume every planar graph with less than n vertices can be 6-colored Assume G has n vertices Since G is planar, it has some

node v with degree at most 5 Remove v and color by Induction Hypothesis Not too difficult to give an inductive proof of 5-colorability, using same fact that some vertex has degree 5 4-color theorem remains challenging! Implementing Graphs Adjacency Matrix

Suppose we have a graph G with n vertices. The adjacency matrix is the n x n matrix A=[aij] with: aij = 1 if (i,j) is an edge aij = 0 if (i,j) is not an edge Good for dense graphs! Example 0 A= 1 1

1 1 0 1 1 1 1 0 1

1 1 1 0 Counting Paths The number of paths of length k from node i to node j is the entry in position (i,j) in the matrix Ak 0

A2 = 1 1 1 1 0 1 1 1 1

0 1 1 1 1 0 3 = 2 2

2 2 3 2 2 2 2 3 2

2 2 2 3 0 1 1 1

1 0 1 1 1 1 0 1 1

1 1 0 Adjacency List Suppose we have a graph G with n vertices. The adjacency list is the list that contains all the nodes that each node is adjacent to Good for sparse graphs!

Example 1 3 2 4 1:

2: 3: 4: 2,3 1,3,4 1,2,4 2,3 Trees Counting Trees

Different Characterizations Planar Graphs Definition Eulers Theorem Coloring Planar Graphs Heres What You Need to Know

Adjacency Matrix and List Definition