(6 points) Does there exist a digraph D in which no two vertices of D have...
2. (5 Points) Given adjacency list representation of a digraph below with 10 vertices from 0 to 9, does it have a topological order? If so, provide one. Otherwise, explain why. 0: 4 2 1 3 1: 2 2: 3 3: 4: 2 5: 1 4 3 8 9 6: 3 2 1 9 7: 2 1 8: 2 1 4 6 9: 3 1 4
(6 points) Let D be a digraph in which the sum of all of the outdegrees is 26. (a) What must the sum of all of the indegrees be? (b) How many arcs does this digraph have?
topic: graph theory
Question 5. Prove that every graph with at least two vertices contains two vertices with the same degree. Then for each n 2 2 give an example of a graph with n vertices which does not have three vertices of the same degree.
Question 5. Prove that every graph with at least two vertices contains two vertices with the same degree. Then for each n 2 2 give an example of a graph with n vertices which...
Does there exist a set of intervals, no 5 of which share a point, such that the interval graph (this is the graph formed by taking the vertices to be the intervals, and then you connect two of the vertices by an edge if the corresponding intervals intersect) is non-planar? Prove or disprove. Please do not just give the definition of interval graphs as others have for this same question.
A bipartite graph is a graph in which the vertices can be divided into two disjoint nonempty sets A and B such that no two vertices in A are adjacent and no two vertices in B are adjacent. The complete bipartite graph Km,n is a bipartite graph in which |A| = m and |B| = n, and every vertex in A is adjacent to every vertex in B. (a) Sketch K3,2. (b) How many edges does Km,n have? (c) For...
Prove that every graph with two or more nodes must have at least two vertices having the same degree. Determine all graphs that contain just a single pair of vertices that have exactly the same degree.
4) Which does not exist as an electron sublevel? a) 2d 6) 44 c) Sd d) 75
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...
Let S = {n ∈ N | 1 ≤ n < 6} and R = {(m, n) ∈ S × S | m ≡ n mod 3} a. List all numbers of S. b. List all ordered pairs in R. c. Does R satisfy any of the following properties: (R), (AR), (S), (AS), and/or (T)? d. Draw the digraph D presenting the relation R where S are the vertices, and R determines the directed edges. e. Give each edge in...
Question 16. A maximal plane
graph is a plane graph G = (V, E) with n ≥ 3 vertices such that if
we join any two non-adjacent vertices in G, we obtain a non-plane
graph. (a) Draw a maximal plane graphs on six vertices. (b) Show
that a maximal plane graph on n points has 3n − 6 edges and 2n − 4
faces. (c) A triangulation of an n-gon is a plane graph whose
infinite face boundary is a...