b please The statement of Ore's Theorem is:If G is a graph of order n ≥ 3 such that for all pairs of distinct non-adjacentvertices x and y, deg x + degy ≥ n, then G is Hamiltonian.The full strength of the proof of Ore's theorem was not needed to prove this result.Use either the statement or arguments from the proof to prove the following:(a) If G is a graph of order n ≥ 3 such that 8(G) ≥ n/2 then G is Hamiltonian.(b) Let x and y be distinct nonadjacent vertices of a graph G of order n such thatdeg x + deg y ≥ n. Then G+ xy is Hamiltonian iff G is Hamiltonian.

Answered: Image /qna-images/answer/5eddfa8b-038f-4e96-a0b4-5329fb00df6c.jpg

Answered: The statement of Ore's Theorem is: If G…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 10EQ

See similar textbooks

Similar questions

9. Which of the following statements are true? T(G) is connected for every connected graph G. If T (G) has more than one vertex then it has a cycle. If G and G'are isomorphic then so are T (G) and T(G'). □ IfT(G) and T (G') are isomorphic then G and G' are isomorphic
7. [10 marks] Let G = (V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a cycle in G on which x, y, and z all lie. (a) First prove that there are two internally disjoint xy-paths Po and P₁. (b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that there are three paths Qo, Q1, and Q2 such that: ⚫each Qi starts at z; • each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are distinct; the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex 2) and are disjoint from the paths Po and P₁ (except at the end vertices wo, W1, and w₂). (c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and z all lie. (To do this, notice that two of the w; must be on the same Pj.)
1. (a) For the invariant theory connected to the general linear model, find g. (b) Show that ğ : 0₁ → 9₁ and 9: 0₂ ā (c) Show that 8² = 0 ₂ for all g. Σ7_₁ ? is the maximal invariant in © = {§1, ..., Ék, 0²}.
An independent set of a graph G is a subset I of the vertex set V such that no twovertices in I are adjacent. Let i(G) be the size of a maximal independent set of G.(a) Show that I is an independent set of G if and only if V − I is a vertex coverof G.(b) Conclude from part (a) that i(G) + vc(G) = |V |.
Determine the largest positive integer k such that χ(H) = χ(G) = k, where H is obtained from a nonempty graph G by subdividing each edge of G exactly once.
Prove that the curves o_1={|z| =2} and o_2= {|z - i|=5} a are homotopic.
The following question are mostly on the topic of paths, cycles and connectedness - (a) Let G be a graph of order n ≥ 2 such that (G) ≥ 1/1/2(n − 1). Show that any two non-adjacent vertices in G have a common neighbor. (b) Let G be an (n,m) graph such that m> (21). Show that G is connected. (c) Let G be a connected graph that is not complete. Show that there exists three vertices u, v, w in G such that uv = E(G), vw = E(G), but uw & E(G).
The parts (a) and (b) of this problem are independentof each other.G1 G24 51 236sx yt u v(a) Prove that the graphs G1 and G2 are isomorphic byexhibiting an isomorphism from one to the other byconcrete arguments and verify it by using adjacencymatrices. Please take the ordering of the vertices as1, 2, 3, 4, 5, 6 while forming AG1, adjacency matrix ofG1.Warning: One must stick to the labelings ofthe vertices as they are given, if one changesthe labelings/orderings etc., the solution willnot be taken into account.(b) Consider the complete graph K13 with vertex setV13 = {u1, u2, u3, · · · , u13}.Let H = (V, E) be the simple graph obtained fromK13 by adding a new vertex u, i.e. V = V13 ∪ {u}and deleting the edges {u1, u2} and {u2, u3} andadding the edges {u1, u} and {u, u2} and keepingthe remaining edges same.Determine whether H has an Euler circuit or not,an Euler path or not. One must validate any conclusion by clear arguments.
Prove that every graph G on {1,...n} which is acyclic and has n – 2 edges has exactly 2 connected components.
Manuben
The exercise is as followed: "Let H be the smallest class that contains the graphs K′5 and K′3,3, and is closed under isomorphism and 2-sum. Prove that every graph in H is planar." I think I know how to do the 2-sum of K'5 and K'3,3 (as you can find attached) but I don't know whether it's correct and what I got to do now? Thank you very much for the help!!
b) For in R", we say that is in the kernel of Tif T(x) 0. If x₁ and 2 are both in the kernel of T, show that ax₁ + bx2 is also in the kernel of T for all scalars a and b. =

SEE MORE QUESTIONS

Recommended textbooks for you

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS