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5) What is the sum of all node degrees, in a graph of 7 nodes connected...
An eccentricity of a node in a connected graph G is the maximum distance from a node in G. A node is central in G when its eccentricity is minimum. Show that a tree has either one or two central nodes. Hint: The eccentricity of a node in a tree it its maximum distance from some leaf. Suppose all the leaves of a tree are removed (assuming some node(s) remains after this surgery): will this affect the set of central...
PROMPT: Consider a graph G. A connected component is a maximal subset of nodes that induces a connected sub graph. It’s maximal in the sense that you cannot add a node with the resulting induced sub graph remaining connected.The following function numComponents returns the number of connected components in an undirected graph. QUESTION: What is the time complexity for this function? The time complexity should be a function of the number of nodes |V| and the number of edges |E|....
A graph consists of a collection of nodes (the dots in the figure) connected by edges (line segments from one node to another). A move on a graph is a move from one node to another along a single edge. Find the probability of going from Start to Finish in a sequence of two random moves in the graph shown. Start Finish
Answer all the BLANKS from A to N please. 7. For the graph shown below at the bottom, answer the following questions a) Is the graph directed or undirected? b) What is the deg ()? c) Is the graph connected or unconnected? If it is not connected, give an example of why not d) ls the graph below an example of a wheel? e) Any multiple edges? 0 What is the deg'(E)? ) What is the deg (B)? h) Is...
4. Given a network of 8 nodes and the distance between each node as shown in Figure 1: 4 1 7 0 4 4 6 6 Figure 1: Network graph of 8 nodes a) Find the shortest path tree of node 1 to all the other nodes (node 0, 2, 3, 4, 5, 6 and 7) using Dijkstra's algorithm. b) Design the Matlab code to implement Dijkstra's algorithm 4. Given a network of 8 nodes and the distance between each...
Discrete Mathematics 6: A: Draw a graph with 5 vertices and the requisite number of edges to show that if four of the vertices have degree 2, it would be impossible for the 5 vertex to have degree 1. Repetition of edges is not permitted. (There may not be two different bridges connecting the same pair of vertices.) B: Draw a graph with 4 vertices and determine the largest number of edges the graph can have, assuming repetition of edges...
Recall the definition of the degree of a vertex in a graph. a) Suppose a graph has 7 vertices, each of degree 2 or 3. Is the graph necessarily connected ? b) Now the graph has 7 vertices, each degree 3 or 4. Is it necessarily connected? My professor gave an example in class. He said triangle and a square are graph which are not connected yet each vertex has degree 2. (Paul Zeitz, The Art and Craft of Problem...
PYTHON ONLY Implement the Dijkstra’s Shortest path algorithm in Python. A graph with 10 nodes (Node 0 to node 9) must be implemented. You are supposed to denote the distance of the edges via an adjacency matrix (You can assume the edge weights are either 0 or a positive value). The adjacency matrix is supposed to be a 2-D array and it is to be inputted to the graph. Remember that the adjacency list denotes the edge values for the...
We now consider undirected graphs. Recall that such a graph is • connected iff for all pairs of nodes u, w, there is a path of edges between u and w; • acyclic iff for all pairs of nodes u, w, whenever there is an edge between u and w then there is no path Given an acyclic undirected graph G with n nodes (where n ≥ 1) and a edges, your task is to prove that a ≤ n...
Problem 6. In lecture, we saw that an undirected graph with n nodes can have at most n(n - 1)/2 edges. Such a graph necessarily has one connected component. The greatest number of edges possible in a disconnected graph, however, is smaller. Suppose that G (V, E) is a disconnected graph with n nodes, how large can |El possibly be? You do not need to prove your answer, but you should provide some explanation of how you obtained it.