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Questions and Answers of
Linear Algebra
(a) Determine P(G, λ) for G = K1,3. b) For n e Z+, what is the chromatic polynomial for K1,n? What is its chromatic number?
(a) Consider the graph K2,3 shown in Fig. 11.91, and let λ ˆˆ Z+ denote the number of colors available to properly color the vertices of K2,3. (i) How many proper colorings of K2,3 have vertices
Find the chromatic number of the following graphs.(a) The complete bipartite graphs Km,n.(b) A cycle on n vertices, n ¥ 3.(c) The graphs in Figs. 11.59(d), 11.62(a), and 11.85.(d) The
If G is a loop-free undirected graph with at least one edge, prove that G is bipartite if an only if x (G) = 2.
(a) Determine the chromatic polynomials for the graphs in Fig. 11.92(b) Find x (G) for each graph.(c) If five colors are available, in how many ways can the vertices of each graph be properly colored?
Let G be a loop-free undirected graph on n vertices. If G has 56 edges and has 80 edges, what is n?
If G = (V, E) is an undirected graph, a subset D of V is called a dominating set if for all v ∈ V, either v ∈ D or v is adjacent to a vertex in D. If D is a dominating set and no proper subset of
Let G = (V, E) be the undirected connected "ladder graph" shown in Fig. 11.94. For n ¥ 0, let an denote the number of ways one can select n of the edges in G so that no two edges share a
Consider the four comb graphs in parts (i), (ii), (iii), and (iv) of Fig. 11.96. These graphs have 1 tooth, 2 teeth, 3 teeth, and ft teeth, respectively. For n ¥ 1, let an count the number
Consider the four graphs in parts (i), (ii), (iii), and (iv) of Fig. 11.97. If an counts the number of independent subsets of {x1, x2,....,xn, y1, y2, . . . . , yn}, where n ¥ 1, find and
For n ¥ 1, let the number of edges in Kn, and let a0 = 0. Find the generating function f(x) = n=0 anxn.
For the graph G in Fig. 11.98, answer the following questions.(a) What are γ(G), β(G), and x(G)?(b) Does G have an Euler circuit or a Hamilton cycle?(c) Is G bipartite? Is it
(a) Suppose that the complete bipartite graph Km,n contains 16 edges and satisfies m ≤ n. Determine ra, n so that Km,n possesses (i) An Euler circuit but not a Hamilton cycle; (ii) Both a Hamilton
If G = (V, E) is an undirected loop-free graph, the line graph of G, denoted L(G), is a graph with the set E as vertices, where we join two vertices e1, e2 in L(G) if and only if e1, e2 are adjacent
Explain why each of the following polynomials in λ cannot be a chromatic polynomial. (a) λ4 - 5λ3 + 7λ2 - 6λ + 3 (b) 3λ3 - 4λ2 + λ (c) λ4 - 3λ3 + 5λ2 - 4λ
Determine the number of cycles of length 4 in the hypercube Qn.
(a) For all x, y e Z+, prove that x3y - xy3 is even. (b) Let V = {1,2,3,...,8, 9}. Construct the loop-free undirected graph G = (V, E) as follows: For m, n e V, m ≠ n, draw the edge [m, n] in G if
(a) For n ≥ 1, let Pn-1 denote the path made up of n vertices and n - 1 edges. Let an be the number of independent subsets of vertices in Pn-1. (The empty subset is considered one of these
Suppose that G = (V, E) is a loop-free undirected graph. If G is 5-regular and |V| = 10, prove that G is nonplanar.
(a) If the edges of K6 are painted either red or blue, prove that there is a red triangle or a blue triangle that is a subgraph. (b) Prove that in any group of six people there must be three who are
(a) Let G = (V, E) be a loop-free undirected graph. Recall that G is called self-complementary if G and are isomorphic. If G is self-complementary (i) determine |E| if |V| = n; (ii) prove that G is
(a) Show that the graphs G1 and G2, in Fig. 11.95, are isomorphic.(b) How many different isomorphisms f: G1 G2 are possible here?
Are any of the planar graphs for the five Platonic solids bipartite?
(a) How many paths of length 5 are there in the complete bipartite graph K37? (Remember that a path such as v1 → v2 → v3 → v4 → v5 → v6 is considered to be the same as the path v6 → v5
Let x = {1, 2, 3, ... , n}, where n ≥ 2. Construct the loop- free undirected graph G = (V, E) as follows: • (V): Each two-element subset of X determines a vertex of G. • (E): If v1, v2 ∈ V
If G = (V, E) is an undirected graph, a subset K of V is called a covering of G if for every edge {a, b} of G either a or b is in K. The set K is a minimal covering if K - {x} fails to cover G for
(a) Draw the graphs of all nonisomorphic trees on six vertices. (b) How many isomers does hexane (C6H14) have?
Find two nonisomorphic spanning trees for the complete bipartite graph K2,3. How many nonisomorphic spanning trees are there for K2,3?
For n ∈ Z+, how many nonisomorphic spanning trees are there for K2,n?
For each graph in Fig. 12.7, determine how many nonidentical (though some may be isomorphic) spanning trees exist.
Let T = (V, E) be a tree where |V| = n. Suppose that for each v ∈ V, deg(v) = 1 or deg(v) ≥ m, where m is a fixed positive integer and m ≥ 2.(a) What is the smallest value possible for n?(b)
Let G = (V, E) be a loop-free connected undirected graph. Let H be a subgraph of G. The complement of H in G is the subgraph of G made up of those edges in G that are not in H (along with the
Complete the proof of Theorem 12.5. The following statements are equivalent for a loop-free undirected graph G = (V, E). (a) G is connected, and |V| = |E| + 1. (b) G contains no cycles, and if a, b
A labeled tree is one wherein the vertices are labeled. If the tree has n vertices, then {1, 2, 3,..., n} is used as the set of labels. We find that two trees that are isomorphic without labels may
Let n ∈ Z+, n ≥ 3. If v is a vertex in Kn, how many of the nn-2 spanning trees of Kn have v as a pendant vertex?
Characterize the trees whose Prufer codes (a) Contain only one integer, or (b) Have distinct integers in all positions.
Show that the number of labeled trees with n vertices, k of which are pendant vertices, is (n - k)!S(n - 2, n - k) = (n!/k!)S(n - 2, n - k), where S(n - 2, n - k) is a Stirling number of the second
Let G = (V, E) be the undirected graph in Fig. 12.9. Show that the edge set E can be partitioned as E1 ª E2 so that the subgraphs G1 = (V, E1), G2 = (V, E2) are isomorphic spanning trees
(a) Let F1 = (V1, E1) be a forest of seven trees where |E1| = 40. What is |V1|? (b) If F2 = (V2, E2) is a forest with |V2| = 62 and |E2| = 51, how many trees determine F2?
(a) Verify that all trees are planar. (b) Derive Theorem 12.3 from part (a) and Euler's Theorem for planar graphs.
Give an example of an undirected graph G = (V, E) where |V| = |E| + 1 but G is not a tree.
(a) If a tree has four vertices of degree 2, one vertex of degree 3, two of degree 4, and one of degree 5, how many pendant vertices does it have? (b) If a tree T = (V, E) has v2 vertices of degree
If G = (V, E) is a loop-free undirected graph, prove that G is a tree if there is a unique path between any two vertices of G.
Answer the following questions for the tree shown in Fig. 12.28.(a) Which vertices are the leaves? (b) Which vertex is the root? (c) Which vertex is the parent of g? (d) Which vertices are the
Prove Theorem 12.6 and Corollary 12.1. Theorem 12.6 Let T = (V, E) be a complete m-ary tree with |V| = n. If T has i leaves and i internal vertices, then (a) n = mi + 1; (b) ℓ = (m - 1)i + 1;
With m, n, i, ℓ as in Theorem 12.6, prove that (a) n = (mℓ - 1 )/(m - 1). (b) ℓ = [(m - 1)n + 1/m.
(a) A complete ternary (or 3-ary) tree T = (V, E) has 34 internal vertices. How many edges does T have? How many leaves? (b) How many internal vertices does a complete 5-ary tree with 817 leaves have?
The complete binary tree T = (V, E) has V = {a, b, c, ...,i, j, k}. The post order listing of V yields d, e, b, h, i, f, j, k, g, c, a. From this information draw T if (a) The height of T is 3; (b)
For m ¥ 3, a complete m-ary tree can be transformed into a complete binary tree by applying the idea shown in Fig. 12.32.(a) Use this technique to transform the complete ternary decision
(a) At a men's singles tennis tournament, each of 25 players brings a can of tennis balls. When a match is played, one can of balls is opened and used, then kept by the loser. The winner takes the
Use a complete ternary decision tree to repeat Example 12.15 for a set of 12 coins, exactly one of which is heavier (and counterfeit).
Let T = (V, E) be a binary tree. In Fig. 12.29 we find the subtree of T rooted at vertex p. (The dashed line coming into vertex p indicates that there is more to the tree T than what appears in the
Let T = (V, E) be a balanced complete m-ary tree of height h ≥ 2. If T has ℓ leaves and bh-1 internal vertices at level h - 1, explain why ℓ = mh-1 + (m - 1)bh-1.
Consider the complete binary trees on 31 vertices. (Here we distinguish left from right as in Example 12.9.) How many of these trees have 11 vertices in the left subtree of the root? How many have 21
Consider the following algorithm where the input is a rooted tree with root r.Step 1: Push r onto the (empty) stackStep 2: While the stack is not emptyPop the vertex at the top of the stack and
Consider the following algorithm where the input is a rooted tree with root r.Step 1: Push r onto the (empty) stackStep 2: While the stack is not emptyIf the entry at the top of the stack is not
Let T = (V, E) be a rooted tree ordered by a universal address system, (a) If vertex v in T has address 2.1.3.6, what is the smallest number of siblings that v must have? (b) For the vertex v in part
For the tree shown in Fig. 12.30, list the vertices according to a preorder traversal, an inorder traversal, and a postorder traversal.
List the vertices in the tree shown in Fig. 12.31 when they are visited in a preorder traversal and in a postorder traversal.
(a) Find the depth-first spanning tree for the graph shown in Fig. 11.72(a) if the order of the vertices is given as (i) a, b, c, d, e, f, g, h; (ii) h, g, f, e, d, c, b, a; (iii) a, b, c, d, h, g,
Find the breadth-first spanning trees for the graphs and prescribed orders given in Exercise 7. (a) Find the depth-first spanning tree for the graph shown in Fig. 11.72(a) if the order of the
Let G = (V, E) be an undirected graph with adjacency matrix A(G) as shown here.Use a breadth-first search based on A(G) to determine whether G is connected.
(a) Give an example of two lists L1, L2, each of which is in ascending order and contains five elements, and where nine comparisons are needed to merge L1, L2 by the algorithm given in Lemma
Apply the merge sort to each of the following lists. Draw the splitting and merging trees for each application of the procedure. (a) - 1, 0, 2, - 2, 3, 6, - 3, 5, 1, 4 (b) - 1, 7, 4, 11, 5, - 8, 15,
Related to the merge sort is a somewhat more efficient procedure called the quick sort. Here we start with a list L: a1, a2, . . . , an, and use a1 as a pivot to develop two sublists L1 and L2 as
Prove that the function g used in the second method to analyze the (worst-case) time-complexity of the merge sort is monotone increasing.
Construct an optimal prefix code for the symbols a, b, c, . . . , i, j that occur (in a given sample) with respective frequencies 78, 16, 30, 35, 125, 31, 20, 50, 80, 3.
Let T = (V, E) be a complete m-ary tree of height h. This tree is called a full m-ary tree if all of its leaves are at level h. If T is a full m-ary tree with height 7 and 279,936 leaves, how many
Let T be a full m-ary tree with height h and v vertices. Determine h in terms of m and v.
Using the weights 2, 3, 5, 10, 10, show that the height of a Huffman tree for a given set of weights is not unique. How would you modify the algorithm so as to always produce a Huffman tree of
Let Ll, for 1 ≤ i ≤ 4, be four lists of numbers, each sorted in ascending order. The numbers of entries in these lists are 75, 40, 110, and 50, respectively. (a) How many comparisons are needed
Find the articulation points and biconnected components for the graph shown in Fig. 12.44.
Let G = (V, E) be a loop-free connected undirected graph, where V = {a, b, c, . . . , h, i, j}. Ordering the vertices alphabetically, the depth-first spanning tree T for G - with a as the root-is
In step (2) of the algorithm for articulation points, is it really necessary to compute low(x1) and low(x2)?
Let G = (V, E) be a loop-free connected undirected graph with v ˆˆ V.(b) If v is an articulation point of G, prove that v cannot be an articulation point of .
If G = (V, E) is a loop-free undirected graph, we call G color-critical if X(G - v) < X(G) for all v ∈ V. (We examined such graphs earlier, in Exercise 19 of Section 11.6.) Prove that a
Does the result in Lemma 12.4 remain true if T = (V, E') is a breadth-first spanning tree for G = (V, E)?
Prove Lemma 12.3. Let G = (V, E) be a loop-free connected undirected graph with z ∈ V. The vertex z is an articulation point of G if and only if there exist distinct x, y ∈ V with x ≠ z, y ≠
Let T = (V, E) be a tree with |V| = n ≥ 3. (a) What are the smallest and the largest numbers of articulation points that T can have? Describe the trees for each of these cases. (b) How many
(a) Let T = (V, E) be a tree. If v ∈ V, prove that v is an articulation point of T if and only if deg(v) > 1. (b) Let G = (V, E) be a loop-free connected undirected graph with |E| ≥ 1. Prove that
Let G = (V, E) be a loop-free connected undirected graph with |V| ≥ 3. If G has no articulation points, prove that G has no pendant vertices.
For the loop-free connected undirected graph G in Fig. 12.43(i), order the vertices alphabetically.(a) Determine the depth-first spanning tree T for G with e as the root. (b) Apply the algorithm
Answer the questions posed in the previous exercise but this time order the vertices as h, g, f, e, d, c, b, a and let c be the root of T. (a) Determine the depth-first spanning tree T for G with e
Let G = (V, E) be a loop-free undirected graph with |V| = n. Prove that G is a tree if and only if P(G, λ) = λ(λ - 1)n-1.
(a) Let T = {V, E) be a complete 6-ary tree of height 8. If T is balanced, but not full, determine the minimum and maximum values for |V|. (b) Answer part (a) if T = (V, E) is a complete m-ary tree
The rooted Fibonacci trees Tn, n ¥ 1, are defined recursively as follows:(1) T1 is the rooted tree consisting of only the root;(2) T2 is the same as T1 - it too is a rooted tree that
(a) The graph in part (a) of Fig. 12.48 has exactly one spanning tree - namely, the graph itself. The graph in Fig. 12.48(b) has four nonidentical, though isomorphic, spanning trees. In part (c) of
Let G = (V, E) be the undirected connected "ladder graph" shown in Fig. 12.49. For n ¥ 0, let an count the number of spanning trees of G, whereas bn counts the number of these spanning
Let T = (V, E) be a tree where |V| = v and |E| = e. The tree T is called graceful if it is possible to assign the labels {1, 2, 3, .. . , v} to the vertices of T in such a manner that the induced
For an undirected graph G = (V, E) a subset of I of V is called independent when no two vertices in I are adjacent. If, in addition, I ª {x} is not independent for each x V -
In part (i) of Fig. 12.51 we find a graceful labeling of the caterpillar shown in part (i) of Fig. 12.50. Find a graceful labeling for the caterpillars in part (ii) of Figs. 12.50 and 12.51.
Develop an algorithm to gracefully label the vertices of a caterpillar with at least two edges.
Consider the caterpillar in part (i) of Fig. 12.50. If we label each edge of the spine with a 1 and each of the other edges with a 0, the caterpillar can be represented by a binary string. Here that
For n ¥ 0, we want to count the number of ordered rooted trees on n + 1 vertices. The five trees in Fig. 12.52(a) cover the case for n = 3.(a) Performing a postorder traversal of each tree
A telephone communication system is set up at a company where 125 executives are employed. The system is initialized by the president, who calls her four vice presidents. Each vice president then
For n ¥ 1, let tn count the number of spanning trees for the fan on n + 1 vertices. The fan for n = 4 is shown in Fig. 12.53.(a) Show that tn+1 = tn + nl=0 tl, where n
(a) Consider the subgraph of G (in Fig. 12.54) induced by the vertices a, b, c, d. This graph is called a kite. How many nonidentical (though some may be isomorphic) spanning trees are there for this
Let T be a complete binary tree with the vertices of T ordered by a preorder traversal. This traversal assigns the label 1 to all internal vertices of T and the label 0 to each leaf. The sequence of
For k ∈ Z+, let n = 2k, and consider the list L: a1, a2, a3, ..., an. To sort L in ascending order, first compare the entries at and al+(n/2), for each 1 ≤ i ≤ n/2. For the resulting 2k-1
Let G = (V, E) be a loop-free undirected graph. If deg(v) ≥ 2 for all v ∈ V, prove that G contains a cycle.
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