Homework Answers
Let n >k> 1 with n even and k odd. Make a k-regular graph G by putting n vertices in a circle and...
need help with a and b in this graph theory question
Let n >k> 1 with n even and k odd. Make a k-regular graph G by putting n vertices in a circle and connecting each vertex to the exact a) Show that for all u,v there are k internally disjoint u, v-paths (you (b) Use the previous part, even if you did not prove it, to show that the e vertex and the k 1 closest vertices on either...
7. Write g(x) in vertex form. Then sketch the graph of g(x) g(x) = 2x +...
7. Write g(x) in vertex form. Then sketch the graph of g(x) g(x) = 2x + 4x - 6. Show ALL work a) Write the equation in vertex form. b) Suate the vertex -- c) State the axis-of-symmetry. a) Find the intercepts. X-intercept(s): y-intercept(s); 8. Given h(x) = x(x + 2)*(x - 2)? (1) Determine the degree and end behavior. Degree As X- As x 0 (b) Find the x-intercepts, the multiplicity of each root, and state whether the graph...
Problem 1. Consider a graph G with n vertices: • Ifq is the size of the...
Problem 1. Consider a graph G with n vertices: • Ifq is the size of the largest independent set in graph G, show that ax(G) n. • Use the previous result to prove that x(G)(n - d) > n and thuse x(G) > n/(n - d), where d is the minimal degree of the vertex in G. • Prove that x(G) + X(G) Sn +1 (Hint: Induction). • x(G)x(G) 2 n. • Use previous inequality to show that x(G) +...
Question 15: Let n 〉 r 〉 1 . Prove that any n-vertex graph of minimum degree more than n -n/r contains Kr+1 without u...
Question 15: Let n 〉 r 〉 1 . Prove that any n-vertex graph of minimum degree more than n -n/r contains Kr+1 without using Turán's Theorem.
Question 15: Let n 〉 r 〉 1 . Prove that any n-vertex graph of minimum degree more than n -n/r contains Kr+1 without using Turán's Theorem.
graph G, let Bi(G) max{IS|: SC V(G) and Vu, v E S, d(u, v) 2 i},...
graph G, let Bi(G) max{IS|: SC V(G) and Vu, v E S, d(u, v) 2 i}, 10. (7 points) Given a where d(u, v) is the length of a shortest path between u and v. (a) (0.5 point) What is B1(G)? (b) (1.5 points) Let Pn be the path with n vertices. What is B;(Pn)? (c) (2 points) Show that if G is an n-vertex 3-regular graph, then B2(G) < . Further- more, find a 3-regular graph H such that...
Graph theory a) prove that there is a 3 regular graph if n is even and...
Graph theory
a) prove that there is a 3 regular graph if n is even and
n>=4
b)
thank you
Exercise 4. How many distinct graphs are there with vertex set {1,2,...,n}? How many distinct graphs are there with vertex set {1,2,...,n} and m edges? For these questions, what happens if "with vertex set {1,2,...,n}" is replaced by "with n vertices"?
* Exercise 1: Let G be the graph with vertex set V(G) = Zi,-{0,-, that two...
* Exercise 1: Let G be the graph with vertex set V(G) = Zi,-{0,-, that two vertices x, y E V(G) are connected by an edge if and only if ,10) and such ryt5 mod 11 or xEy t7 mod 11 1. Draw the graph G. 2. Show that the graph G is Eulerian, i.e., it has a closed trail containing all its edges
a. A minimal verter in a directed graph is a vertex v such that there are no edges (u, ) n the graph for any u. Argue that if the resource allocation graph G = (PUR, E) has a minimal vertex p E P the...
a. A minimal verter in a directed graph is a vertex v such that there are no edges (u, ) n the graph for any u. Argue that if the resource allocation graph G = (PUR, E) has a minimal vertex p E P then the system can make progress 1 mark
a. A minimal verter in a directed graph is a vertex v such that there are no edges (u, ) n the graph for any u. Argue that...
8, (10 pts) Show that given a directed graph G = (V,E) already stored in adjacency matrix form, determining if there is a vertex with in-degree n - 1 and out-degree 0 can be done in O(n) time whe...
8, (10 pts) Show that given a directed graph G = (V,E) already stored in adjacency matrix form, determining if there is a vertex with in-degree n - 1 and out-degree 0 can be done in O(n) time where n is the number of vertices in V.
8, (10 pts) Show that given a directed graph G = (V,E) already stored in adjacency matrix form, determining if there is a vertex with in-degree n - 1 and out-degree 0 can...
Question 4. For n 2, let Gn be the grid graph, whose vertex set is V={(x, y) E Z × Z : 0 < x < n,0
topic: graph theory
Question 4. For n 2, let Gn be the grid graph, whose vertex set is V={(x, y) E Z × Z : 0 < x < n,0
need help with a and b in this graph theory question
Let n >k> 1 with n even and k odd. Make a k-regular graph G by putting n vertices in a circle and connecting each vertex to the exact a) Show that for all u,v there are k internally disjoint u, v-paths (you (b) Use the previous part, even if you did not prove it, to show that the e vertex and the k 1 closest vertices on either...
7. Write g(x) in vertex form. Then sketch the graph of g(x) g(x) = 2x + 4x - 6. Show ALL work a) Write the equation in vertex form. b) Suate the vertex -- c) State the axis-of-symmetry. a) Find the intercepts. X-intercept(s): y-intercept(s); 8. Given h(x) = x(x + 2)*(x - 2)? (1) Determine the degree and end behavior. Degree As X- As x 0 (b) Find the x-intercepts, the multiplicity of each root, and state whether the graph...
Problem 1. Consider a graph G with n vertices: • Ifq is the size of the largest independent set in graph G, show that ax(G) n. • Use the previous result to prove that x(G)(n - d) > n and thuse x(G) > n/(n - d), where d is the minimal degree of the vertex in G. • Prove that x(G) + X(G) Sn +1 (Hint: Induction). • x(G)x(G) 2 n. • Use previous inequality to show that x(G) +...
Question 15: Let n 〉 r 〉 1 . Prove that any n-vertex graph of minimum degree more than n -n/r contains Kr+1 without using Turán's Theorem.
Question 15: Let n 〉 r 〉 1 . Prove that any n-vertex graph of minimum degree more than n -n/r contains Kr+1 without using Turán's Theorem.
graph G, let Bi(G) max{IS|: SC V(G) and Vu, v E S, d(u, v) 2 i}, 10. (7 points) Given a where d(u, v) is the length of a shortest path between u and v. (a) (0.5 point) What is B1(G)? (b) (1.5 points) Let Pn be the path with n vertices. What is B;(Pn)? (c) (2 points) Show that if G is an n-vertex 3-regular graph, then B2(G) < . Further- more, find a 3-regular graph H such that...
Graph theory
a) prove that there is a 3 regular graph if n is even and
n>=4
b)
thank you
Exercise 4. How many distinct graphs are there with vertex set {1,2,...,n}? How many distinct graphs are there with vertex set {1,2,...,n} and m edges? For these questions, what happens if "with vertex set {1,2,...,n}" is replaced by "with n vertices"?
* Exercise 1: Let G be the graph with vertex set V(G) = Zi,-{0,-, that two vertices x, y E V(G) are connected by an edge if and only if ,10) and such ryt5 mod 11 or xEy t7 mod 11 1. Draw the graph G. 2. Show that the graph G is Eulerian, i.e., it has a closed trail containing all its edges
a. A minimal verter in a directed graph is a vertex v such that there are no edges (u, ) n the graph for any u. Argue that if the resource allocation graph G = (PUR, E) has a minimal vertex p E P then the system can make progress 1 mark
a. A minimal verter in a directed graph is a vertex v such that there are no edges (u, ) n the graph for any u. Argue that...
8, (10 pts) Show that given a directed graph G = (V,E) already stored in adjacency matrix form, determining if there is a vertex with in-degree n - 1 and out-degree 0 can be done in O(n) time where n is the number of vertices in V.
8, (10 pts) Show that given a directed graph G = (V,E) already stored in adjacency matrix form, determining if there is a vertex with in-degree n - 1 and out-degree 0 can...
topic: graph theory
Question 4. For n 2, let Gn be the grid graph, whose vertex set is V={(x, y) E Z × Z : 0 < x < n,0
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