Let G be an undirected graph with n vertices. Let A(G) be the maximum degree of any vertex in G, 8(G) be the minimum degree of any vertex in G, and m be the number of edges in G. Prove that: [8(G)*n]/2 <= m <= [A(G)*2]/2

Q: Let G be a simple undirected graph (without multiple edges and loops) with  n vertices and m edges.…

A:

Q: 5. Let G = (V, E) be a finite, connected graph. Show that if no vertex of G has degree less than 2,…

A:

Q: Let G be a simple plane graph with fewer than 12 faces, in which each vertex has degree at least 3.…

A: Step 1: Consider the provided question, (17) (i)

Q: Let G be a graph with n 1 vertices. (a) If x(G) = 1, what can you say about E(G), the edge set of G?…

A: Given : G is a graph with n≥1 vertices We have to answer the following : (a) If χG=1, what can be…

Q: A simple graph G on n vertices with m edges is overfull if m> [n/2]A. Show that if G is overfull,…

A: Given that: A simple graph G on n vertices with m edges is overfull if m&gt;n2△. The objective is to…

Q: Let G be a simple graph on 9 vertices, and assume we know that the sum of all degrees in G is at…

A:

Q: Prove that every connected graph with m vertices has at least (m-1) edges, for every m>1

A:

Q: When two graphs G and H are isomorphic, this means: there exists a bijection between the vertices of…

A: Two graphs G and H are said to be isomorphic, if there exist a bijection f between their vertices…

Q: Prove that having n vertices, where n is a positive integer, is an invariant for graph isomorphism.

A:

Q: Suppose a graph G is regular of degree r, where r is odd. (a) prove that G has an even even number…

A:

Q: Let G be a connected cubic graph of order n > 4 having clique number 3. Determine x(G).

A: Given- G be a connected cubic graph of order n&gt;4 having clique number 3. To Find- The value of χG…

Q: Let G be a connected graph with at least one edge and F ⊆ E(G) be an edge cut. Prove that F is a…

A:

Q: 18. Let G be a graph with n vertices and exactly n-1 edges . Prove that G has either a vertex of…

A: Now, we need to prove that if G is a graph with n vertices and (n-1) edges then G has either a…

Q: Let G be a connected graph with n(G) ≥ 2. Show that G at least 2 vertices which are not cut-vertices…

A:

Q: Prove that if the graph G is discontinuous, then the complementary graph G is continuous.

A:

Q: Prove that every graph G with n vertices and chromatic number k = x(G) has at most · (n2 – ) edges.…

A:

Q: Let Gbe a simple graph with at least 11l vertices, and let G be its complement. (i) Prove that Gand…

A: Given: G is a simple graph with at least 11 vertices and Ḡ be its complement.

Q: Let G be a connected simple graph in which all vertices have degree 4. Prove that it is possible to…

A:

Q: Q10. Using Corollary 3.1.24 of the book, to prove that a graph G is bipartite if and only if a(H) =…

A: Given: α(H)=β'(H) To find: Isolated vertices

Q: 2. Give an example of a graph with at least four vertices, or prove that none exists, such that: (a)…

A: The objective is to give an example of a graph with at leas four vertices, or prove that none…

Q: Prove that if u is a vertex of odd degree in a graph, then there exists a path from u to another…

A: We have to prove :       If u is a vertex of odd degree in a graph, then there exists a path from…

Q: 15. Let G be a graph on n vertices (with no loops or multiple edges). Prove that if all vertices in…

A: 1. A graph G is an ordered pair (V(G), E(G)) consisting of a non-empty set V(G) of vertices and a…

Q: Find a subgraph H of the graph G given below that has 4 vertices and 3 edges.

A:

Q: Find the smallest 3-regular graph G such that K(G) = 1.

A:

Q: Let G = (V, E) be a k-connected graph, let æ E V, and let B C V\{x} such that |B| > k. Show that…

A: Fan: Let v∈V(G) and let A∈V(G)\{v}. A v,A-fan of size k is a collection of k paths P1,P2,...,Pk so…

Q: If J has at least one vertex of degree greater than zero then there are two of even degree as long…

A: This is a problem of Graph Theory.

Q: Let G be a cubic graph on 100 vertices. Determine the diameter of the complement of G.

A:

Q: Let G be a connected graph of order n = 4 and let k be an integer with 2 ≤ k ≤ n − 2. a)Prove that…

A:

Q: c) Prove that if a plane graph G has n vertices, e edges, f faces and k components then n- e+f =k+1.

A: We can prove Euler's formula (n-e+ f = k+1) works generally for anyconnected simple planar graph.so,

Q: For any connected graph G of order 55, if x is a vertex of G, then the number of components of G-x…

A: A connected graph is graph that is connected in the sense of a topological space, i.e., there is a…

Q: State whether the following statement is true or false. Justify your answer with a short proof or a…

A: Consider the general case, Let G be k-edge connected graph then we shall rove that G is k connected.…

Q: Prove that every simple graph G= (V, E) with |V] > 2 has at least two vertices of equal degree. %3D…

A: We have a simple graph, with V≥2, that is, the number of vertices is greater than or equal to 2.…

Q: If G has at least one vertex of degree greater than zero then there are two of even degree as long…

A: This is the problem of graph theory.

Q: Show that it is not possible to creak graph the" degree with g vertices such that every a of vertex…

A: Given that  The no of vertices = 9 degree of each vertices =3 To show such graph is not possible.

Q: 5. Let G be a connected graph without loops and of order at least 11. Prove that G or its complement…

A:

Q: Let H be a connected planar graph with at least 3 vertices. Prove that f is greater than or equal to…

A:

Q: Let G be a connected graph of order n = 4 and let k be an integer with 2 ≤ k ≤ n − 2. Prove that if…

A:

Q: Let G be a graph with exactly 10 vertices and 27 edges. Suppose that each vertex has degree 3, 5, or…

A:

Q: 5.1.21. Suppose that every edge of a graph G appears in at most one cycle. Prove that every block of…

A:

Q: Let G be a simple graph with 20 vertices. Suppose that G h ponents, and every pair of distinct…

A: Given: Simple graph G with 20 vertices. G has two-component with distinct vertices that satisfy…

Q: Let G be a graph with v vertices and e edges. Let M be the maximum degree of the vertices of G, and…

A: We are given a G graph G with v number of vertices and e number of edges in the graph. we are also…

Q: 5. Let G be a graph with n vertices, t of which have degree k and the others have degree k+1. Prove…

A:

Q: For any connected graph G of order 54, if x is a vertex of G, then the number of components of G-x…

A: Given: To explain the given statement as follows,

Q: Let G be a graph on at least 3 vertices. Suppose that G has the property that G – v is connected for…

A: If we are confident with our definition we can easily attempt like these questions, G is connected…

Q: Let G be a simple graph with nonadjacent vertices v and w, and let G+e denote the simple graph…

A: Given that, v and w are non-adjacent vertices in G. (1): G+e the simple graph formed by adding the…

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!