QUESTION 1: Given the following adjacency matrix representation of a graph to answer the followed questions. 01 1 0 0 0 0 1 1 1 0 1 |0 1 1 1 1 1 1 A. is it directed graph or not and why? B. How many vertices are there in the graph? C. How many edges are there in the graph? D. reRepresent the graph using an adjacency list. E. Draw the original graph

QUESTION 1: Given the following adjacency matrix representation of a graph to answer the followed questions. 01 1 0 0 0 0 1 1 1 0 1 |0 1 1 1 1 1 1 A. is it directed graph or not and why? B. How many vertices are there in the graph? C. How many edges are there in the graph? D. reRepresent the graph using an adjacency list. E. Draw the original graph

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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I really want to learn. Please explain too. For each graph representation, what is the appropriate worst-case time complexity for checking if two distinct vertices are connected. The choices are: O(1), O(V), O(E), or O(V+E) Adjancy Matrix = ____ Edge List = ____ Adjacency List = ____
Consider the un-directed graph shown in the Fig. The values inside the node refer to the feature value of the node. The order of the nodes is A,B,C,D when following the matrix form in below questions.(You can use python or other languages to find the eigen values and other operationfor this question):1. For the given graph, write down the degree matrix, adjacency matrix, and Laplacian matrix representation.2.Find the eigenvalues and eigenvectors of the graph Laplacian matrix. Do you see any pattern pertaining to the signs of the eigenvalues (all positive or negative or no such pattern). Will it be same for all such graph Laplacianmatrices, and why?
8. Answer the following questions for a random graph generated by the Erdos-Renyi random graph model ER(n, p). You do not need to prove your answers are correct but still need to show your work. Your answer should be in terms of n and p. 8(a) What is the expected number of triangles? 8(b) What is the probability that d(u, v) > 2 for two particular vertices u v? (Hint: First, find the probability that d(u, v) > 1. Then consider, for all other vertices x, the probability that uxv is NOT a path in the random graph. Multiply all these probabilities together.) Note that for this question, no graph is given. However, we know that the expected number of edges for ER(n, p) random graph model is , and that the expected degrees of the vertex for the random graph model ER(n, p) is (n-1) *p
An Euler path, in a graph or multigraph, is a walk through the graph which uses every edge exactly once. An Euler circuit is an Euler path which starts and stops at the same vertex. NOTE: graphs are in the image attached. Which of the graphs below have Euler paths? Which have Euler circuits? List the degrees of each vertex of the graphs above. Is there a connection between degrees and the existence of Euler paths and circuits? Is it possible for a graph with a degree 1 vertex to have an Euler circuit? If so, draw one. If not, explain why not. What about an Euler path? What if every vertex of the graph has degree 2. Is there an Euler path? An Euler circuit? Draw some graphs. Below is part of a graph. Even though you can only see some of the vertices, can you deduce whether the graph will have an Euler path or circuit? NOTE: graphs is in the image attached.
a) Illustrate the sequence of vertices of this graph visited using depth-first search traversal starting at vertex g. b) Illustrate adjacency list representation and adjacency matrix representation, respectively, for this graph. What are the advantages and disadvantages of those two representations? c) Describe an algorithm to find in the graph a path illustrated below that goes through every edge exactly once in each direction.
2. Let G = (V, E) be a directed weighted graph with the vertices V = {A, B, C, D, E, F) and the edges E= {(A, B, 12), (A, D, 17), (B, C, 8), (B, D, 13), (B, E, 15), (B, F, 13), (C, E, 12), (C, F, 25)}, where the third components is the cost. (a) Write down the adjacency list representation the graph G = (V, E).
Question-1 : For the following situation: (a) Represent it with adjacency list representation. (b) According to your representation in part- a, is it a sparse graph or dense graph(c) What is the memory requirement for this case? (d) Represent the part-c in general asymptotic notation (e) If you have represented it in adjacent matrix representation, what will be your worst time complexity for a quick edge search time? [Hints: you can write it in a single word like O (answer).] (f) If you have represented it in adjacent matrix representation, will it be a Sparse or Dense graph? Why? What's your logic?
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Given a vertex set W = {1, 2, 3, 4}, solve for the following questions: (i) Knowing that edges are the same if and only if they have the same endpoint, how many different edges are possible? (ii) Suppose for this question that graphs are different if they have different sets of edges (they do not depend on if the graphs are isomorphic or not). How many simple graphs are there with the vertex set W? (iii) Draw the eleven nonisomorphic simple graphs that have four vertices.
Consider the graph depicted below. Answer the following questions: What is the degree of vertex a? What is the sum of degrees of all vertices? Who are the neighbors of vertex f? [Please, give the list in alphabetical order, separating letters with only a comma. Do not insert blank spaces] How many zeros does the adjacency matrix of the graph above contain? What is the value on row g and column d of the adjacency matrix?
1. For each pair of vertex in no. 3 give the vertex sequence of the longest path connecting them that repeat no edges. Is there a longest path connecting them? 2. How many spanning trees are can be generated from its graph? Define 2 spanning trees of Graph A and B. 3. Which graph has and Euler path? Euler Circuit? (Use Fleury’s Algorithm) For those that don’t give an explanation? 4. Which graph contains Hamiltonian Path? Hamiltonian Circuit? For those that don’t give an explanation?
(4) For the following Directed Graph, create an adjacency matrix. Then, via matrix multiplication, give me a COMPLETE list of all pairs of nodes that can reach one another in THREE moves. (Keep in mind when listing an ordered pair, the node you start from goes first, and the node you end on goes last.)