GRAPH THEORY and APPLICATIONS
▫ Tree: a connected graph with no cycle (acyclic). ▫ Forest: a graph with no cycle. ▫ Paths are trees. ▫ Star: A tree consisting of one vertex adjacent to.
Chapter 10.1 Trees
Trees. Prof. Tesler. Math 184A. Winter 2017. Prof. Tesler. Ch. 10.1: Trees. Math 184A / Winter 2017. 1 / 15. Page 2. Trees. Tree in graph theory. Stick figure
Module 8: Trees and Graphs - Theme 1
Theorem 1. A tree with n nodes has n -1 edges. Proof. Every node except the root has exactly one in-coming edge. Since there
4. Trees
The following result shows the existence of spanning trees in connected graphs. Theorem 4.12 Every connected graph has at least one spanning tree. Proof Let G
GRAPHS AND TREES
Thus each edge of a directed graph can be drawn as an arrow going from the first vertex to the second vertex of the ordered pair. Graphs: Definitions and Basic
An Introduction to Combinatorics and Graph Theory
available in this pdf file. . w1 . w2 . w3 . w4 . w5 . w6 . w7 . v1 . v2 In general spanning trees are not unique
Introduction to Graph Theory
1.1s. Write down the number of vertices the number of edges
1.10. Trees and Spanning Trees. 1.10.1. Definition: Tree. ∗ ∗ ∗
3. Page 6. WUCT121. Graphs. 54. When the graph has a large number of vertices it is not easy to find all spanning trees. In fact
An introduction to chordal graphs and clique trees
Clique trees and chordal graphs have carved out a niche for themselves in recent work on sparse matrix algorithms due primarily to research questions
3.1 Characterizations and Properties of Trees 3.2 Rooted Trees
D. Page 7. GRAPH THEORY – LECTURE 4: TREES. 7. Lemma 1.10. Let v and w be two vertices in a tree T such that w is of maximum distance from v (i.e. ecc(v) =
Chapter 10.1 Trees
Trees. Tree in graph theory. Stick figure tree. Not a tree. (has cycle). Not a tree. (not connected). A tree is an undirected connected graph with no cycles
GRAPH THEORY and APPLICATIONS
? Tree: a connected graph with no cycle (acyclic). ? Forest: a graph with no cycle. ? Paths are trees. ? Star: A tree consisting of one vertex adjacent to.
GRAPHS AND TREES
Note that each directed graph has an associated ordinary. (undirected) graph which is obtained by ignoring the directions of the edges. Graphs: Definitions and
3.1 Characterizations and Properties of Trees 3.2 Rooted Trees
GRAPH THEORY – LECTURE 4: TREES. Abstract. Review from §1.5 tree = connected graph with no cycles. Def 1.1. In an undirected tree a leaf is a vertex of ...
Introduction to Graph Theory
examples of graphs connectedness
An Introduction to Combinatorics and Graph Theory
Graph theory is concerned with various types of networks new tree by introducing a new root vertex and making the children of this root the two.
GRAPH THEORY WITH APPLICATIONS
12.2 The Number of Spanning Trees. Applications. 12.3 Perfect Squares . Appendix I Hints to Starred Exercises. Appendix II Four Graphs and a Table of their
Partitions of Graphs into Trees
Maximal planar bipartite graphs have a 2-tree partition as shown by Ringel [14]. Here we give a different proof of this result with a linear time algorithm.
Chapter 6: Graph Theory
Rather than finding a minimum spanning tree that visits every vertex of a graph an Euler path or circuit can be used to find a way to visit every edge of a
Graphs and Trees
Lots of terminology surrounding graphs tons of types
Graph Theory III - MIT - Massachusetts Institute of Technology
Grow a tree one edge at a time by adding the minimum weight edge of the graph tothe tree making sure that you have a tree at each step ALG2: Select edges one at a time always choosing the minimum weight edge that does notcreate a cycle with previously selected edges
Graph Theory: Trees - IIT Kgp
GRAPH THEORY { LECTURE 4: TREES Abstract x3 1 presents some standard characterizations and properties of trees x3 2 presents several di erent types of trees x3 7 develops a counting method based on a bijection between labeled trees and numeric strings x3 8 showns how binary trees can be counted by the Catalan recursion Outline
Introduction to graph theory - University of Oxford
Trees and forests A tree (a connected acyclic graph) A forest (a graph with tree components) ©Department of Psychology University of Melbourne Bipartite graphs A bipartite graph (vertex set can be partitioned into 2 subsets and there are no edges linking vertices in the same set) A complete bipartite graph (all possible edges are present) K1
Graph Theory I - Properties of Trees
3 Trees Definition 4Given a graph G • A path in G is a sequence of edges such that each edge begins where the previous edge ends and ends where the next edge begins • A cycle in G is a path starting and ending at the same vertex G is called a tree if it contains no cycles
Lecture 12: Introduction to Graphs and Trees
Binary search tree (BST) - a tree where nodes are organized in a sorted order to make it easier to search At every node you are guaranteed: All nodes rooted at the le† child are smaller than the current node value All nodes rooted at the right child are smaller than the current node value
Searches related to trees in graph theory pdf PDF
GRAPH THEORY { LECTURE 5: SPANNING TREES 3 Choosing a Frontier Edge Def 1 3 Let T be a tree subgraph of a graph G and let S be the set of frontier edges for T The function nextEdge(GS) (usually deterministic) chooses and returns as its value the frontier edge in S that is to be added to tree T Def 1 4
What is the difference between a tree and a spanning tree?
Trees and Spanning Trees •A graph having no cycles is acyclic. •A forest is an acyclic graph. •A leaf is a vertex of degree 1. •A spanning sub-graph of G is a sub-graph with vertex set V(G). •A spanning tree is a spanning sub-graph that is a tree.
What is a graph in physics?
Definition 1AgraphGis a setV(G)of points (called vertices) together with a setE(G)of edges connectingthe vertices. Though graphs are abstract objects, they are very naturally represented by diagrams, where we (usually)draw the vertices and edges in the plane.
What is a tree acyclic structure of linked nodes?
Tree- a directed, acyclic structure of linked nodesNode- an object containing a data value and links to other nodes All the blue circles Tree- a directed, acyclic structure of linked nodesEdge- directed link, representing relationships between nodes All the grey lines
How do you implement a graph search algorithm?
Goal: implement the basic graph search algorithms in timeO(m+n). This is linear time, since it takesO(m+n)time simply to read the input. Note that when we work with connected graphs, a running time ofO(m+n)is the same asO(m), sincemn 1. Breadth First Search (BFS)Depth First Search (DFS) Example… Start at the start. Look at all the neighbors.
GRAPH THEORY
and APPLICATIONSTrees Graph Theory and Applications © 2007 A. Yayimli2PropertiesTree: a connected graph with no cycle (acyclic)
Forest: a graph with no cycle
Pathsare trees.
Star: A tree consisting of one vertex adjacent to
all the others. Graph Theory and Applications © 2007 A. Yayimli3Trees as ModelsTrees are used in many applications:
analysis of algorithms, compilation of algebraic expressions, theoretical models of computation, etc.Search trees
Abstract models: sort techniques like heapsort.
Graph Theory and Applications © 2007 A. Yayimli4Number of EdgesTheorem: In every tree T = (V, E), |V| = |E| + 1Proof: by induction on number of edges.
If |E| = 0 then the tree consists of a single isolated vertex. |V| = 1 = |E| + 1 Assume that the theorem is true for trees of at most k edges.Graph Theory and Applications © 2007 A. Yayimli5Number of EdgesConsider a tree T, where |E| = k + 1
T he edge (y,z) is removed: Two subtrees T 1 and T 2 |V| = |V 1 | + |V 2 | and |E| = |E 1 | + |E 2 | + 1.Since 0 |E
1 | k and 0 |E 2 | k, |V 1 | = |E 1 | + 1 and |V 2 | = |E 2 | + 1.Consequently,
|V| = |V 1 | + |V 2 | = |E 1 | + 1 + |E 2 | + 1 = |E 1 | + |E 2 | + 1 + 1 = |E| + 1 y zGraph Theory and Applications © 2007 A. Yayimli6Definition of TreeTheorem: The following statements are equivalent
for a loop-free undirected graph G = (V, E): A.G is a tree.
B. G is connected, but the removal of any edge from G disconnects G into two subgraphs that are trees. C.G contains no cycle, and |V| = |E| + 1.
D.G is connected, and |V| = |E| + 1.
E.G contains no cycle, and if a, b ӇV with
(a, b) ӈE, then the graph obtained by adding (a, b) to G has precisely one cycle. Graph Theory and Applications © 2007 A. Yayimli7ProofA ҧB
If G is a tree, then G is connected.
Let e = (a,b) be any edge of G. Then, ifG-e is connected, there are at least two paths in G from a to b.From two such paths we can form a cycle.
But G has no cycle.
Hence, G-e is disconnected and the vertices in G-e can be partitioned into two subsets: Vertex a and those vertices that can be reached from a. Vertex b and those vertices that can be reached from b. These two connected components are trees, because a loop or cycle in either component would also be in G. Graph Theory and Applications © 2007 A. Yayimli8Proof (cont.)B ҧC
If G contains a cycle then let e = (a,b) be an edge of the cycle.But then, G-e is connected, contradicting the
hypothesis in part B.So, G contains no cycle.
Since G is a loop-free connected graph, we know thatG is a tree.
Then, |V| = |E| + 1.
Graph Theory and Applications © 2007 A. Yayimli9Proof (cont.)C ҧD
Assume G is not connected.
Let G 1 , G 2 , ..., G r be components of G.For 1 i r, select a vertex v
i ӇG i and add the r-1 edges (v 1 ,v 2 ), (v 2 ,v 3 ), ..., (v r-1 ,v r ) to G to form G'.G' is a tree.
|V| = |E'| + 1From C, |V| = |E| + 1, so |E| = |E'| and r-1 = 0
With r = 1, it follows that G is connected.
To complete the proof:
D ҧE ӟEҧA
Graph Theory and Applications © 2007 A. Yayimli10More Definitions on Graphs Distance: If G has a (u,v) path, then the distance from u to v, d(u,v) is the least length of a (u,v) path.EccentricityǪ(u) of a vertex u is max
v˝V d(u,v).The radiusof a graph G is min
u˝VǪ(u)
4 4 4 4
23 3 44
4 3 4 3
Each vertex is labeled
with its eccentricity.Radius: 2
Diameter: 4 (maximum
eccentricity) Graph Theory and Applications © 2007 A. Yayimli11Center of a TreeCenter: The subgraph induced by the vertices of
minimum eccentricity.Theorem: The center of a tree is a vertex or an
edge (two adjacent vertices).5 4 4
5 4 3 3 4 5
5 4 4 5
Graph Theory and Applications © 2007 A. Yayimli12BranchA branch at a nodeu of a tree is a maximal
subtree containing u as an endnode.The number of branches at u is the degree of u.
The weight at a nodeu is the maximum number of
edges in any branch at u. u Graph Theory and Applications © 2007 A. Yayimli13Centroid of a TreeA node v is a centroid, if v has minimum weight.
The centroid of a treeconsist of all centroid
nodes.Theorem: The centroid of a tree is a vertex or an
edge (two adjacent vertices). 1112 6 9 12
Graph Theory and Applications © 2007 A. Yayimli14Center CentroidSmallest trees with one and two central and
centroid nodes:1 CENTER 2
1 CENT- ROID 2 Graph Theory and Applications © 2007 A. Yayimli15Wiener IndexIn a communication network, large diameter is
acceptable if most pairs can communicate via shortest paths.We study the average distance
Average: Sum divided by n(n-1)/2. (all pairs)
It is equivalent to study:
This sum is called the Wiener Indexof G.
G uvVDG d uv
Graph Theory and Applications © 2007 A. Yayimli16Directed TreeEdges of a tree may be directed.
If is a directed edge, then:
u is the parentof v, v is the childof u. A vertex v is the rootof a directed tree, if there are paths from v to every other vertex in the tree. Graph Theory and Applications © 2007 A. Yayimli17Rooted TreeDefinition: A rooted treeis a tree in which we
identify a vertex v as root (indegree: 0).Levelof a vertex: Vertices at distance i from the
root lie at level i+1.The heightof the rooted tree is its maximum
level. root Graph Theory and Applications © 2007 A. Yayimli18Ordered TreesOrdered tree: A directed tree in which the set of
children of each vertex is ordered.Binary Tree: An ordered tree in which no vertex
has more than two children:Left child and right child.
Complete binary tree: Every vertex has either
two children or none.Balancedcomplete binary tree: Every endpoint
(leaf) has the same level. Graph Theory and Applications © 2007 A. Yayimli19Complete TreesTheorem:A complete balanced binary tree of height h has
2 h - 1 vertices.A complete balanced N-ary tree of height h has
vertices. 1 1 h N N Graph Theory and Applications © 2007 A. Yayimli20Cut EdgeA cut edgeof G is an edge e such that G - e is
disconnected.This graph has 3 cut edges.Theorem: A connected graph is a tree if and only
if every edge is a cut edge. Graph Theory and Applications © 2007 A. Yayimli21Edge CutFor subsets S and S' of V,
[S,S'] is the set of edges with one end in S, the other in S'.Edge cut: A subset of E of the form [S,S'], where
S is a nonempty proper subset of V,
S' = V - S.
Bond: A minimal nonempty edge cut of G.
If G is connected, then a bond B is a minimal subset of E such that G - B is disconnected. an edge cut a bond Graph Theory and Applications © 2007 A. Yayimli22Cut VertexA vertex v is a cut vertex if:
E can be partitioned into two nonempty subsets E
1 and E 2 G[E 1 ] and G[E 2 ] have just the vertex v in common. cut vertices Graph Theory and Applications © 2007 A. Yayimli23Spanning TreeA spanning tree of a connected undirected
graph G is a subgraph which is a tree and which contains all the vertices of G.The construction of a communication network
A road map or railway system
Graph Theory and Applications © 2007 A. Yayimli24Minimum-weighted Spanning TreeProblem: Given the cost of directly connecting any
two nodes, problem is to find a network: at minimum cost and providing route between every two nodesSolution: The solution is the minimum-weighted
spanning tree of the associated weighted graph.Minimum-weighted spanning tree can be found
by an efficient algorithm. Graph Theory and Applications © 2007 A. Yayimli25Steiner TreeA generalization of the minimum-weighted
spanning tree problem: Given a proper subset V' of the vertices of a graph find a minimum-weighted tree which spans the vertices of V'.Such a tree is called SteinerTree.
No efficient algorithm is known for Steiner tree
problem. Graph Theory and Applications © 2007 A. Yayimli26Enumeration of TreesWith one or two vertices, only one tree can be
formed.With three vertices there is one isomorphism
class. The adjacency matrix is determined by which vertex is the center.So, there are 3 trees with 3 vertices.
Graph Theory and Applications © 2007 A. Yayimli27Enumeration of TreesWith 4 vertices:
There are 4 stars and 12 paths
This yields to 16 trees.
With 5 vertices, a careful study yields 125 trees.Wit vertices, there are n
n-2 trees: this is Cayley's Formula. a b d b c d a c Graph Theory and Applications © 2007 A. Yayimli28Spanning Trees in a GraphThe complete graph wit vertices has all the
edges that can be used in forming trees wit vertices.The number of spanning trees in a complete graph
wit vertices is n n-2Can we find a method to compute the number of
spanning trees in any graph?Not containing
the diagonalContaining
the diagonal Graph Theory and Applications © 2007 A. Yayimli29ContractionDefinition: In a graph G, contractionof edge e
with end points u and v is the replacement of u and v with a single vertex the incident edges of this vertex are the edges other than e that were incident to u and v.The resulting graph G·e has one less edge
than G. u e v GG·e
Graph Theory and Applications © 2007 A. Yayimli30Recursive SolutionProposition: Let ˫(G) denote the number of
spanning trees of a graph G. If e ӇE is not a loop, then:˫(G) = ˫(G-e) + ˫(G·e)
Example:
a b c d e G G - e4 spanning treesG·e
4 spanning trees
Graph Theory and Applications © 2007 A. Yayimli31Recursive SolutionThis may lead to a recursive algorithm.
We cannot apply the recurrence when e is a
loop. The loops do not affect the number of spanning trees.Hence, we can delete loops as they arise.
If we try to compute by deleting and contracting
every edge, the amount of computation grows exponentially with the size of the graph. Graph Theory and Applications © 2007 A. Yayimli32Matrix Tree ComputationForm a matrix:
Put the vertex degrees on the diagonal
The remaining elements are 0.
Substract the adjacency matrix from it.
Example:
2000d0300c0030b0002a d c b a 0110
d1011c1101b0110a d c b a
2-1-10
d -13-1-1 c -1-13-1 b0-1-12
a d c b aMatrix for the Kite
Graph Theory and Applications © 2007 A. Yayimli33Matrix Tree ComputationDelete a row and a column of the resulting
matrix.Take the determinant.
det: -4 +0 + 0 -0 -2 -2 = -8 2-1 -10 d -1 3 -1 -1 c -1-1 3-1 b 0-1 -12 a d c b a 2-10 d -1-1-1 b 0-12 a d c a Graph Theory and Applications © 2007 A. Yayimli34Matrix Tree TheoremGiven a loopless graph G:
Vertex set: v
1 , v 2 , ..., v n Let a ij be the number of edges with endpoints v i and v jLet Q be the matrix in which entry (i,j) is:
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