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 twonon-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 verticesu, v, w in G such that uv = E(G), vw = E(G), but uw & E(G).

(a) Show that any two non-adjacent vertices in G have a common neighbor Given:G is a graph with n…

Answered: The following question are mostly on…

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter10: Analytic Geometry

Section10.CT: Test

Problem 6CT: Using the table from Exercise 5, sketch the graph of 2x+3y=12.

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Using the table from Exercise 5, sketch the graph of 2x+3y=12.
Let Vn be the set of connected graphs having n edges, vertex set [n], and exactly one cycle. Form a graph Gn whose vertex set is Vn. Include {gn, hn} as an edge of Gn if and only if gn and hn differ by two edges, i.e. you can obtain one from the other by moving a single edge. Tell us anything you can about the graph Gn. For example, (a) How many vertices does it have? (b) Is it regular (i.e. all vertices the same degree)? (c) Is it connected? (d) What is its diameter?
Let G be a simple graph with nonadjacent vertices v and w, and let G+e denote the simple graph obtained from G by creating a new edge, e, joining v and w. Prove that x(G) = min{x(G+e), x((G+e) 4e)}.
Let G be a simple graph with 11 vertices, each of degree 5 or 6. Prove that G has at least 7 vertices of degree 8 or at least 6 vertices of degree 7. Do not use the planar equation e <= 3v - 6.
Let G be a graph on at least 3 vertices. Suppose that G has the property that G – v is connected for each vertex v of G. (a) Prove that G must be a connected graph. (b) Prove that every vertex of G has degree at least 2.
At a cupcake stall at the Greenwich market, the owner is selling six different types of cupcakes. The owner sells them in boxes where each box contains exactly two cupcakes, each of different type. Each type of cupcake is used in combination with at least three others. (a) Let G be a graph where each vertex represents one of the cupcakes and where an edge joins two vertices whenever two cupcakes are used in the same box. Let v be an arbitrary vertex in G. What is the smallest degree v can have? Justify your answer. ' ^ (b) Show that there are three gift boxes which between them have all six types of cupcakes.
Let G be a graph of order n that is isomorphic to its complement G. How many edges does G have? Explain your answer. If a graph G has n vertices, all of which but one have odd degree, how many vertices of odd degree are there in G, the complement of G? Prove your answer.
2n 4. Prove or disprove: for every n-vertex graph G, x(G) ≤ 2w(G) + a(G) +10.
How to do b
Show that For n > 1 let Gn be the simple graph with vertex set V(Gn) = {1,2, ., n} in which two different vertices i and j are adjacent whenever j is a multiple of i or i is a multiple of j. For what n is Gn planar? ...1
Let G be a graph and e € E(G). Let H be the graph with V(H) = V(G) and E(H) = E(G) \ {e}. Then e is a bridge of G if H has a greater number of connected components than G. = (a) Let G be the simple graph with V(G) {u, v, w, x, y, z) and E(G) {uy, vx, vz, wx, xz}. For each e € E(G), state whethere is a bridge of G. Justify your answer.
6. Determine if the following statement is true or false. If it is true, then prove it. If it is false, give a counter example. Let G be a connected graph of order n ≥ 3. If u and v are adjacent vertices of G, then |e(u) - e(v)| ≤ 1.

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