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mathematics
statistics

Questions and Answers of Statistics

If the simple graph G has v vertices and e edges, how many edges does have?
If the degree sequence of the simple graph G is d1, d2, . . . , dn, what is the degree sequence of G?
Show that if G is a simple graph with n vertices, then the union of G and is Kn.
Draw the converse of each of the graphs in Exercises 7-9 in Section 10.1.
In Exercise determine the number of vertices and edges and find the in-degree and out-degree of each vertex for the given directed multi-graph.
Draw the mesh network for interconnecting nine parallel processors.
Show that every pair of processors in a mesh network of n = m2 processors can communicate using O (√n) = O (m) hops between directly connected processors.
In Exercise use an adjacency list to represent the given graph.
In Exercise draw a graph with the given adjacency matrix.
In Exercise represent the given graph using an adjacency matrix.
In Exercise draw an undirected graph represented by the given adjacency matrix.
In Exercise find the adjacency matrix of the given directed multi graph with respect to the vertices listed in alphabetic order.
In Exercise draw the graph represented by the given adjacency matrix.
Is every zero-one square matrix that is symmetric and has zeros on the diagonal the adjacency matrix of a simple graph?
Use an incidence matrix to represent the graphs in Exercises 13-15.In Exercise 13-15
What is the sum of the entries in a column of the adjacency matrix for an undirected graph? For a directed graph?
What is the sum of the entries in a column of the incidence matrix for an undirected graph?
Find incidence matrices for the graphs in parts (a)-(d) of Exercise 32. In Exercise 32 a) Kn b) Cn c) Wn d) Km,n e) Qn
In Exercise determine whether the given pair of graphs is isomorphic. Exhibit an isomorphism or provide a rigorous argument that none exists.
Show that isomorphism of simple graphs is an equivalence relation.
Show that the vertices of a bipartite graph with two or more vertices can be ordered so that its adjacency matrix has the formWhere the four entries shown are rectangular blocks a simple graph G is
Represent the graph in Exercise 1 with an adjacency matrix.In Exercise 1
Find a self-complementary simple graph with five vertices.
For which integer's n is Cn self-complementary?
How many non isomorphic simple graphs are there with five vertices and three edges?
Are the simple graphs with the following adjacency matrices isomorphic?(a)(b) (c)
Extend the definition of isomorphism of simple graphs to undirected graphs containing loops and multiple edges.
In Exercise determine whether the given pair of directed graphs is isomorphic.
Show that if G and H are isomorphic directed graphs, then the converses of G and H (defined in the preamble of Exercise 67 of Section 10.2) are also isomorphic.
Find a pair of non isomorphic graphs with the same degree sequence such that one graph is bipartite, but the other graph is not bipartite.
What is the product of the incidence matrix and its transpose for an undirected graph?
Represent the graph in Exercise 3 with an adjacency matrix.In Exercise 3
Find a devil's pair for the test that checks the degree sequence in two graphs to make sure they agree.
Represent each of these graphs with an adjacency matrix. a) K4 b) K1,4 c) K2,3 d) C4 e) W4 f) Q3
Does each of these lists of vertices form a path in the following graph? Which paths are simple? Which are circuits? What are the lengths of those that are paths?a) a, e, b, c, bb) a, e, a, d, b, c,
Determine whether each of these graphs is strongly connected and if not, whether it is weakly connected.(a)(b) (c)
What do the strongly connected components of a telephone call graph represent?
Find the strongly connected components of each of these graphs.(a)(b) (c) Suppose that G = (V, E) is a directed graph. A vertex w ˆˆ V is reachable from a vertex v ˆˆ V if there
Show that if G = (V, E) is a directed graph, and then the strong components of two vertices u and v of V are either the same or disjoint.
Find the number of paths of length n between two different vertices in K4 if n is a) 2. b) 3. c) 4. d) 5.
Use paths either to show that these graphs are not isomorphic or to find an isomorphism between them.
Find the number of paths of length n between any two nonadjacent vertices in K3, 3 for the values of n in Exercise 19. In Exercise 19 a) 2. b) 3. c) 4. d) 5.
Find the number of paths from a to e in the directed graph in Exercise 2 of length a) 2. b) 3. c) 4. d) 5. e) 6. f) 7.
Let G = (V, E) be a simple graph. Let R be the relation on V consisting of pairs of vertices (u, v) such that there is a path from u to v or such that u = v. Show that R is an equivalence relation.
In Exercise find all the cut vertices of the given graph.
Suppose that v is an endpoint of a cut edge. Prove that v is a cut vertex if and only if this vertex is not pendant.
Show that a simple graph with at least two vertices has at least two vertices that are not cut vertices.
A communications link in a network should be provided with a backup link if its failure makes it impossible for some message to be sent. For each of the communications networks shown here in (a) and
What is the significance of a vertex basis in an influence graph find a vertex basis in the influence graph in that example.
Show that if a simple graph G has k connected components and these components have n1, n2, . . . , nk vertices, respectively, then the number of edges of G does not exceed
Show that a simple graph G with n vertices is connected if it has more than (n − 1) (n − 2)/2 edges.
How many non isomorphic connected simple graphs are there with n vertices when n is a) 2? b) 3? c) 4? d) 5?
Show that each of the graphs in Exercise 48 has no cut edges. In Exercise 48 a) Cn where n ≥ 3 b) Wn where n ≥ 3 c) Km,n where m ≥ 2 and n ≥ 2 d) Qn where n ≥ 2
Show that if G is a connected graph, then it is possible to remove vertices to disconnect G if and only if G is not a complete graph
Find K(Km,n) and λ(Km,n), where m and n are positive integers.
Let P1 and P2 be two simple paths between the vertices u and v in the simple graph G that do not contain the same set of edges. Show that there is a simple circuit in G.
Explain how Theorem 2 can be used to determine whether a graph is connected
Show that a simple graph G is bipartite if and only if it has no circuits with an odd number of edges.
Use a graph model and a path in your graph, as in Exercise 64, to solve the jealous husband's problem. Two married couples, each a husband and a wife want to cross a river. They can only use a boat
What do the connected components of acquaintanceship graphs represent?
Explain why in the collaboration graph of mathematicians a vertex representing a mathematician is in the same connected component as the vertex representing Paul Erdos if and only if that
When can the centerlines of the streets in a city be painted without traveling a street more than once? (Assume that all the streets are two-way streets.)
In Exercise determine whether the picture shown can be drawn with a pencil in a continuous motion without lifting the pencil or retracing part of the picture.
Show that a directed multi graph having no isolated vertices has an Euler path but not an Euler circuit if and only if the graph is weakly connected and the in-degree and out-degree of each vertex
Devise an algorithm for constructing Euler paths in directed graphs.
For which values of n do the graphs in Exercise 26 have an Euler path but no Euler circuit? In Exercise 26 a) Kn b) Cn c) Wn d) Qn
Find the least number of times it is necessary to lift a pencil from the paper when drawing each of the graphs in Exercises 1-7 without retracing any part of the graph.
In Exercise determine whether the given graph has an Euler circuit. Construct such a circuit when one exists. If no Euler circuit exists, determine whether the graph has an Euler path and construct
In Exercise determine whether the given graph has a Hamilton circuit. If it does, find such a circuit. If it does not, give an argument to show why no such circuit exists.
Does the graph in Exercise 34 have a Hamilton path? If so, find such a path. If it does not, give an argument to show why no such path exists.In Exercise 34
For which values of m and n does the complete bipartite graph Km,n have a Hamilton circuit?
For each of these graphs, determine (i) whether Dirac's theorem can be used to show that the graph has a Hamilton circuit, (ii) whether Ore's theorem can be used to show that the graph has a Hamilton
Show that there is a Gray code of order n whenever n is a positive integer, or equivalently, show that the n-cube Qn, n > 1, always has a Hamilton circuit.
Express Fleury's algorithm in pseudo code. Fleury's algorithm, published in 1883, constructs Euler circuits by first choosing an arbitrary vertex of a connected multi graph, and then forming a
Give a variant of Fleury's algorithm to produce Euler paths. Fleury's algorithm, published in 1883, constructs Euler circuits by first choosing an arbitrary vertex of a connected multi graph, and
Show that a bipartite graph with an odd number of vertices does not have a Hamilton circuit.
Draw the graph that represents the legal moves of a knight on a 3 Ã— 4 chessboard.A knight is a chess piece that can move either two spaces horizontally and one space vertically or one
Show that there is a knight's tour on a 3 Ã— 4 chessboardA knight is a chess piece that can move either two spaces horizontally and one space vertically or one space horizontally and two
Show that there is no knight's tour on a 4 Ã— 4 chessboard.A knight is a chess piece that can move either two spaces horizontally and one space vertically or one space horizontally and
Show that there is no reentrant knight's tour on an m × n chessboard when m and n are both odd.
For each of these problems about a subway system, describe a weighted graph model that can be used to solve the problem. a) What is the least amount of time required to travel between two stops? b)
Find a shortest route (in distance) between computer centers in each of these pairs of cities in the communications network shown in Figure 2.a) Boston and Los Angelesb) New York and San Franciscoc)
Find a least expensive route, in monthly lease charges, between the pairs of computer centers in Exercise 11 using the lease charges given in Figure 2.a) Boston and Los Angelesb) New York and San
The weighted graphs in the figures here show some major roads in New Jersey. Part (a) shows the distances between cities on these roads; part (b) shows the tolls.a) Find a shortest route in distance
What are some applications where it is necessary to find the length of a longest simple path between two vertices in a weighted graph?
Use Floyd's algorithm to find the distance between all pairs of vertices in the weighted graph in Figure 4(a).
Give a big-O estimate of the number of operations (comparisons and additions) used by Floyd's algorithm to determine the shortest distance between every pair of vertices in a weighted simple graph
Solve the traveling salesperson problem for this graph by finding the total weight of all Hamilton circuits and determining a circuit with minimum total weight.
Find a route with the least total airfare that visits each of the cities in this graph, where the weight on an edge is the least price available for a flight between the two cities.
Construct a weighted undirected graph such that the total weight of a circuit that visits every vertex at least once is minimized for a circuit that visits some vertices more than once.
In Exercise find the length of a shortest path between a and z in the given weighted graph.
The longest path problem in a weighted directed graph with no simple circuits asks for a path in this graph such that the sum of its edge weights is a maximum. Devise an algorithm for solving the
Find a shortest path between a and z in each of the weighted graphs in Exercises 2-4.In Exercise 2-4
Find shortest paths in the weighted graph in Exercise 3 between the pairs of vertices in Exercise 6. In Exercise 6 a) a and d b) a and f c) c and f d) b and z
Find a combination of flights with the least total air time between the pairs of cities in Exercise 8, using the flight times shown in Figure 1. In Exercise 8 a) New York and Los Angeles b) Boston
Can five houses be connected to two utilities without connections crossing?
Show that K5 is non planar using an argument similar to that given in Example 3.
Suppose that a connected planar graph has six vertices, each of degree four. Into how many regions is the plane divided by a planar representation of this graph?
Suppose that a connected planar simple graph with e edges and v vertices contains no simple circuits of length 4 or less. Show that e ≤ (5/3)v − (10/3) if v ≥ 4.
Which of these non planar graphs have the property that the removal of any vertex and all edges incident with that vertex produces a planar graph? a) K5 b) K6 c) K3,3 d) K3,4

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