Representation (Matrix) Incidence matrix E x v edge, vertex contains the edge's data Adjacency matrix V x v boolean values (adjacent or not) Or Edge weights Representation (List) Edge list pairs (ordered if directed) of vertices Optionally weight and other data Adjacency list Implementation. Adjacency-list representation an array of v lists, one for each vertex. For each uv, adj u points to all its adjacent vertices. Adjacency lists Advantage: saves space for sparse graphs. Most graphs are sparse. Traverse all the edges that start at v, in (degree(v) Disadvantage: Check for existence of an edge (v, u) in worst case time (degree(v) Adjacency list Storage for a directed graph the number of items are(out-degree (v) e so we need ( v e ). V v v v adjacency matrix representation v x v matrix a ( aij ) such that aij 1 if (i, j ) e and 0 otherwise.

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A graph G contains another graph h if some subgraph of g is h or is isomorphic. H is a proper subgraph if H! G Spanning subgraph Subgraph H has the same vertex set. Possibly not all the edges H spans. Induced Subgraph For any pair of vertices x and y of h, xy is an edge of h if and only if xy is an edge. H has the most edges that appear in g over the same vertex set. Induced Subgraph (2) If h is chosen based on a vertex subset s of V(G then H can be written as gs induced by s a graph that does not contain h as an induced subgraph is said to be h-free component Maximum Connected sub. E., a one-to-one mapping: f : V(G) - v(H) u and v from g are poets adjacent if and only if f(u) and f(v) are adjacent. If an isomorphism can be constructed between two graphs, then we say those graphs are isomorphic. Isomorphism Problem Determining whether two graphs are isomorphic Although these graphs look very different, they are isomorphic; one isomorphism between them is f(a) 1 f(b) 6 f(c) 8 f(d) 3 f(g) 5 f(h) 2 f(i) 4 homework f(j) 7 Graph adt in computer science, a graph. The graph adt follows directly from the graph concept from mathematics.

Sparse/Dense a graph is sparse if e v a graph is dense if e. 2.2.5.5.5 table A weighted graph is a graph for which each edge has an associated weight, usually given by a weight functionw:. Bipartitegraph V can be partitioned into 2 sets V1 and V2such that (u,v)E implies either uV1 and vV2 or vv1 and uV2. Special Types Empty Graph / Edgeless graph no edge null graph no nodes Obviously no edge complete Graph Denoted Kn every pair of vertices are adjacent Has n(n-1) edges Complete bipartite Graph Bipartite variation of Complete Graph every node of one set is connected. Formally, an hypergraph is a pair (X,E) where x is a set of elements, called nodes or vertices, and e is a set of subsets of x, called hyperedges. Hyperedges are arbitrary sets of nodes, contain an arbitrary number of nodes. the degree of b. Degree number of edges incident on a node 1 2 4 5 The in degree of 2 is 2 andthe out degree of 2. Degree (Directed Graphs) In degree: Number of edges entering Out degree: Number of edges leaving Degree indegree outdegree degree: Simple facts If g is a digraph with m edges, then indeg(v) outdeg(v) m e if g is a graph with m edges, then deg(v).

cycle d 1 f e 3 2 Unreachable 4 5 6 Cycle path A path is universities a sequence of vertices such that there is an edge from each vertex to its successor. A path is simple if each vertex is distinct. Simple path from 1 to 5 1, 2, 4, 5 our texts alternates the verticesand edges. If there is path p from u to v then we say v is reachable from u via. cycle d 1 f e 3 2 Unreachable 4 5 6 Cycle cycle a path from a vertex to itself plan is called a cycle. A graph is called cyclic if it contains a cycle; otherwise it is called acyclic Connectivity is connected if you can get from any node to any other by following a sequence of edges or any two nodes are connected by a path. A directed graph is strongly connected if there is a directed path from any node to any other node.

What is a graph? Informally a graph is a set of nodes joined by a set of lines or arrows. Definition: Graph g is an ordered triple g v, e, f) v is a set of nodes, points, or vertices. E is a set, whose elements are known as edges or lines. F is a function maps each element of E to an unordered pair of vertices. Definitions Vertex Basic Element Drawn as a node or a dot. Vertex set of g is usually denoted by V(g or v edge a set of two elements Drawn as a line connecting two vertices, called end vertices, or endpoints. The edge set of g is usually denoted by E(g. Example V:1,2,3,4,5,6 E:1,2,1,5,2,3,2,5,3,4,4,5,4,6 Simple Graphs Simple graphs are graphs without multiple edges or self-loops.

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Presented by mushfiqur rouf (100505056). Graph Theory - history. Leonhard Euler's paper on seven Bridges of Königsberg, published in 1736. The traveling salesman problem. Presentation Transcript, graph Theory - history, leonhard Euler's paper on seven Bridges of Königsberg, published in 1736. The traveling salesman problem, a traveling salesman is to visit a number of cities; how to plan the trip so every city is visited once and just once and the whole trip is as short as possible?

Famous problems, in 1852 Francis Guthrie posed the four color problem which asks if it is possible to color, using only four colors, any map of countries in such a way as to prevent two bordering countries from having the same color. This problem, which was only solved a century later in 1976 by kenneth Appel and Wolfgang haken, can be considered dissertation the birth of graph theory. Cost of wiring electronic components, shortest route between two cities. Shortest distance between all pairs of cities in a road atlas. Matching / Resource Allocation, task scheduling. Visibility / coverage, examples, flow of material liquid flowing through pipes current through electrical networks information through communication networks parts through an assembly line, in Operating systems to model resource handling (deadlock problems). In compilers for parsing and optimizing the code.

Step 3 Draw a line through the intercepts. Example 3 Graph 3x 2y 12 Short cut version Checkpoint. X y 2 Draw quick Graphs Graph the equation. Checkpoint 5x 2y. Draw quick Graphs Graph the equation.

Graphing x #. Graph x -3 The x coordinate is -3 no matter what the value of. Choose any value of y so, the graph of x # is always a vertical line graphing y #. Graph y 2 The y coordinate is 2 no matter what the value of. Choose any value of x so, the graph of y # is always a horizontal line. Graph y -4 The graph of y -4 is horizontal line at -4. Homework: WS: Graphing Equations in Standard Form 1. Introduction to Graph Theory powerPoint Presentation. Download Presentation, introduction to Graph Theory 1 / 54, introduction to Graph Theory.

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Graphing Equations in Standard Form, step 1: Find the x-intercept by letting y 0 and solving for. Step 2: Find the y-intercept by letting x 0 and solving for. Step 3: Draw a line through the two points. Example 3 paper ( ( ) ) 3 2 0 0, let x 0, graph 3x 2y 12 2y 12, let y 0 3x 12 Draw quick Graphs step 1 Find the x-intercept. 3x 12 3 3 x 4 The x-intercept is 4, so plot the point (4, 0). Step 2 Find the y-intercept. 2y 12 2 2 y 6 The y-intercept is 6, so plot the point (0, 6).

Predict the number of CDs purchased by a person who is 27 years old. Find the age of a person who purchased 15 CDs in the previous year. Graph PowerPoint Presentation, download Presentation. Graph 1 /. Find the slope rediff of the line that passes through ( 5, 2) and (-5, -3). 2.4 Part 2 Sept 6, 2013. Graphing Equations in Standard Form Step 1: Find the x-intercept by letting y 0 and solving for. y 3 x 2 3, warm Up, graph. Find the slope of the line that passes through (5, 2) and (-5, -3).

Image/Link below is provided (as is) to download presentation. Download Policy: Content on the website is provided to you as is for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. Presentation Transcript, correlation, used to describe the relationship between the independent variable (x) and the dependent variable (y). Its on a scale from -1 to 1 with the endpoints representing a perfect linear relationship. Practice Problem, find equation of the least-squares line. Find and explain the correlation coefficient,.

Please share this page with them! And if it is helpful to you, remember to like or share. Download, skip this Video, loading SlideShow in 5 Seconds. Solve amp; Graph PowerPoint Presentation, download Presentation. Solve amp; Graph 1 / 16, solve amp; Graph. Graph 3x4y gt; paper -8. Scatterplots amp; Line of Best Fit.

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Please share your experience with presentations in English below. Do you have some useful functional language you could share to help organize a presentation in English? In the lesson, i provided many phrases that are common for native speakers. Ive love to hear from you. You can share in the comments section below. Its the best place to get feedback and learn from the confident English Community. Have a great week! Do you know anyone who needs help with presentations in English?

Augment with. Adapted from umd jimmy lins slides, which is licensed under a creative commons Attribution-Noncommercial. Ilustrace vector infographics set.

Algorithm : Design analysis. In the last class. Implementing Dynamic Set by Union-Find Straight Union-Find.

Adrian Silvescu doina caragea anna Atramentov. The necessity to represent, store and. A graph that compares data by using bars of different lengths and heights. Having the same chance.

Graph, colouring Problem: given a graph, colour. Find the slope of the line that passes through ( 5, 2) and (-5, -3). 2.4 Part 2 Sept.

Algorithms Breadth First search (BFS) Depth First search (DFS) Dijkstras Algorithm. Famous and Productive problem. Lecture 10: Oct. (based on slides in mit.042).