Question

In: Advanced Math

Let G be a graph. For each of the following, determine if the statement is true...

Let G be a graph. For each of the following, determine if the statement is true or false. If it's true, provide a proof and if it's false, provide a counterexample.

(a) G has a cycle if and only if G has a circuit

(b) G has a closed walk if and only if G has a circuit

(c) G has an odd-lengthed cycle if and only if G has an odd-lengthed circuit

(d) G has an odd-lengthed closed walk if and only if G has an odd- lengthed circuit

Solutions

Expert Solution


Related Solutions

Determine whether the following statement about graph theory is true or false. (1) If a graph...
Determine whether the following statement about graph theory is true or false. (1) If a graph with m vertices is connected, then there must be at least m-1 edges. (2) If a graph with m vertices has at least m−1 edges, then the graph must be connected. (3) A simple undirected graph must contain a cycle, if it has m vertices with at least m edges. (4) A graph must contain at least m edges, if it has m vertices...
Let G be a connected graph and let e be a cut edge in G. Let...
Let G be a connected graph and let e be a cut edge in G. Let K be the subgraph of G defined by: V(K) = V(G) and E(K) = E(G) - {e} Prove that K has exactly two connected components. First prove that e cannot be a loop. Thus the endpoint set of e is of the form {v,w}, where v ≠ w. If ṽ∈V(K), prove that either there is a path in K from v to ṽ, or...
Determine whether each statement is true or false. If it is true, prove it. If it...
Determine whether each statement is true or false. If it is true, prove it. If it is false, give a counterexample. a) For every function f : X → Y and all A ⊆ X, we have f^−1 [f[A]] = A. (b) For every function f : X → Y and all A ⊆ X, we have f[X \ A] = Y \ f[A]. (c) For every function f : X → Y and all A, B ⊆ Y ,...
For each of the following statements, determine whether the statement is true or false. If you...
For each of the following statements, determine whether the statement is true or false. If you say the statement is true, explain why and if you say it is false, give an example to illustrate. (a) If {u, v} is a linearly independent set in a vector space V, then the set {2u + 3v, u + v} is also a linear set independent of V. (b) Let A and B be two square matrices of the same format. Then...
For each of the following statements determine if the statement is TRUE, FALSE, or UNCERTAIN. You...
For each of the following statements determine if the statement is TRUE, FALSE, or UNCERTAIN. You must justify your answer graphically. No credit will be given without an explanation. (a) “According to the large-open economy model, if Japan (a large open economy) were to decrease its taxes it would cause both the real interest rate and net exports to increase.” [Hint: In your answer you will need to draw three diagrams] (b) “In the Solow Model with no technological progress,...
Determine whether each of the following statements is true or false. If the statement is false,...
Determine whether each of the following statements is true or false. If the statement is false, modify and rewrite it so that it is a true statement. a. When a molecule has two, degenerate, “infrared active”, vibrational modes, the two vibrational modes will show absorptions at different frequencies in the infrared spectrum. b. For a given substance, strong intermolecular forces between molecules of the substance can cause peak broadening of some of the absorptions in the infrared spectrum of the...
For each of the following statements, determine if the statement is always, sometimes, or never true....
For each of the following statements, determine if the statement is always, sometimes, or never true. Justify your statement with a proof. Hint: To prove that something isn’t always true, it is sufficient to provide a counterexample. (a) If L is an unrecognizable language, then L is (always/sometimes/never) undecidable. (b) If L is a recognizable language then L COMPLEMENT is (always/sometimes/never) recognizable.
Indicate whether each of the following statement is True or False. g) A household purchase of...
Indicate whether each of the following statement is True or False. g) A household purchase of new housing constitutes investment expenditure. h) The larger the marginal propensity to consume the smaller the multiplier. i) Higher consumer confidence in the economy causes a downward movement along aggregate demand curve. j) The higher the interest elasticity of investment the flatter the IS curve. k) The lower the interest sensitivity of money demand the steeper the LM curve. l) To rescue a country’s...
Let G be a simple graph. G is said to be maximal planar if it is...
Let G be a simple graph. G is said to be maximal planar if it is planar and the addition of any new edge to G results in a (simple) non-planar graph. Examples of maximal non-planar graphs are K4 , K5 minus an edge, and K3,3 minus an edge. (a) Show that a maximal planar graph is connected. (b) Show that a maximal planar graph of order ≥3 has no bridges. (c) Show that every face of a maximal planar...
If G = (V, E) is a graph and x ∈ V , let G \...
If G = (V, E) is a graph and x ∈ V , let G \ x be the graph whose vertex set is V \ {x} and whose edges are those edges of G that don’t contain x. Show that every connected finite graph G = (V, E) with at least two vertices has at least two vertices x1, x2 ∈ V such that G \ xi is connected.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT