Suppose G is a simple graph with n >1 vertices such that every vertex has degree at least (n – 1)/2. Prove that G must be connected. (Hint: you may wish to prove this by induction on n).
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Suppose G is a simple graph with n ≥ 1 vertices such that every vertex has degree
at least (n − 1)/2. Prove that G must be connected.
(Hint: you may wish to prove this by induction on n)
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- Let T be a tree of order n and suppose that all vertices of T have degree 1 or degree 3. Prove that T contains exactly n-2/2 vertices of degree 3Prove: “The complement of the graph G is an Eulerian graph whenever G is also an Eulerian graph of odd order and the complement of G is connected.” Give afigure and details to validate the proof.If a simple graph Gis having 14 vertices and 13 edges then its complement graph G will have O 14 vertices and 91 edges O None of these O 78 vertices and 14 edges O 14 vertices and 78 edges
- Draw graph G and its complement, showing that at least one of G and it's complement, G', is connected.4. Let n eN with n > 2 and let Gn = (V, E) be the graph with vertex set V = {v1, v2, ..., v2n} and edge set E = {{v;, Vi+1} : 1Let n, k EN and define the family of graphs G (V, Ek) on the vertex set V,= (1,2,... ,n} and with edge set Enk= {(i,j) EV, x V: i j (mod k)}. In other words, two vertices i and j are adjacent if they are congruent modulok. (a) Draw G42 and G73 and show they are both disconnected. (b) Show that G is connected if and only if k= 1. (c) Show that Gis the empty graph if and only if nRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,