a) Show that if d1, d2, ... , dn is the degree sequence of a tree, then di+d2+ +dn = 2(n-1). b) Find a tree whose degree sequence is 4, 3, 1, 1, 1, 1, 1. c) Is there a tree whose degree sequence is 5, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1? Either exhibit such a tree or prove that no such tree exists. d) Prove or disprove: let d1, d2,..., dn be a sequence of positive integers. There is a tree with degree sequence d1, d2, ...., dn if and only if di + d2 + . .+ dn = 2(n – 1).

Q: Let V be a tree with internal vertices of degree 2, 7, 5, 4., and 6. How many leaves does V have?…

A:

Q: 3.1.8. (!) Prove or disprove: Every tree has at most one perfect matching.

A:

Q: Consider a tournament between N teams, each team playing each of the other teams. 1) How many games…

A: Given information: There is a tournament between N teams, each team playing each of the other teams.

Q: Quèstion 14 For a full binary tree with 10 leaves, the number of vertices are 19. True False A…

A:

Q: 4. Let T be a tree with n ≥ 1 vertices. (a) Show that T has a vertex of degree 1.

A: Given T be a tree with n≥1 vertices. Tree is the non-cyclic connected graph. Tree is the simple…

Q: Let d be the highest degree of any vertex in a tree T, and let l be the number of leaves of T. What…

A:

Q: Determine if (m1,m2,m3,m6) is a spanning sequence of M2×2(R).

A: To determine if m1,m2,m3,m6 is a spanning sequence of M2×2R.

Q: 2. Show that there is no simple graph with six vertices of which the degrees of five vertices are 5,…

A:

Q: (a) Draw a graph with nodes representing the numbers 1, 2, ⋯ , 10, in which two nodes are connected…

A: Here 10 nodes will be present.

Q: 1a. Prove: every tree with n ≥ 2 vertices has at least 2 leaves. (3 pt) 1b. Let T be a tree. Prove:…

A: Question 1a:Let T be a tree with n-vertices, n≥2.  We have to  prove that the tree has at least 2…

Q: (a) If T is a tree with at least 2 vertices, show that x(T) = 2.

A: The given solution is only for part (a) of the question, please resubmit rest of the parts as…

Q: (a) Draw a graph with nodes representing the numbers 1, 2,..., 10, in which two nodes are connected…

A: To find-  Draw a graph with nodes representing the numbers 1, 2,..., 10, in which two nodes are…

Q: 4. (a) (i) List all non-isomorphic trees (not rooted) on 6 vertices with no vertex of degree larger…

A: Consider the given information.   Now, draw all the non-isomorphic tress on 6 vertices with no…

Q: A tree contains some number of leaves (degree I vertices) and four nor

A: Given, a tree contains some number of leaves(degree 1 vertices ) and four non-leaf vertices. The…

Q: Advanced Math Show that if T is a tree on n 2 3 vertices, then the average degree of vertices in T…

A:

Q: Draw an example or prove that there is none A simple digraph with indegrees 0, 1, 2, 2 and…

A:

Q: Show the results of inserting the keys C, R, A, U,J,S, F,T,0,P,M,L,N,W,Q in order into an empty…

A: Given: V={1,2,3,4,5,6,7,8,9}. and edges E(V)={e1=(1,3),e2=(2,4),e3=3,6 ,e4=7,8,…

Q: 1. X = 1+23+74 8- +9/10 5 6 1.1 - SHOW x as an expression tree 1.2- Is the tree binary? 1.3 - Count…

A:

Q: 6. Which of the next sets establish a bipartition of the graph in Fig.1? a) {a, h, g, b, f} and {i,…

A: To determine which of the given sets establish a bipartition of the graph. Here we will use the…

Q: 2 3 4 8 7 The root of the Tree above is node 1. A neighbor of node X is any node which shares an…

A: For the given tree, express the set of all nodes which meet each of the following criteria.

Q: 43.Given that Tree A is bipartite, which of the following gives the grouping of the vertices of Tree…

A:

Q: a) What is the maximum possible order of an element in S,? Give an example of one of these elements.…

A: Given, Group S7 any element can be written in cycle notation i.e. in disjoint cycles as example…

Q: a. How many vertices and how many edges does the pseudograph contain? What is the degree sequence of…

A:

Q: a. Suppose that we have a class with 30 students. Prove that at least two students will have been…

A: a. Suppose that we have a class with 30 students.Prove that at least two students will have been…

Q: |Use the tree defined above to give the subtree rooted at head. Answer:

A: The set of all nodes underneath a particular node x is called the subtree rooted at x.

Q: The total degree of a tree with m vertices is (2m-2). True O False

A: •The degree of a vertex v of a graph G is the number of edges incident on v with self loop…

Q: Let G be a tree with n vertices. Find a formula for the total number of paths G and justify why it…

A:

Q: A sequence of vertices such that each vertex is adjacent to the next vertex is called a(n) a. length…

A: Given question, A sequence of vertices such that each vertex is adjacent to next vertex is called…

Q: A person with $2 in her pocket bets $1, even money,on the flip of a coin, and she continues to bet…

A: Given data :person = $2Her pocket bets $1 , Suppose He bets on head .She gets $1 If geads turn up…

Q: A rooted tree has O a cycles of length one O b. No leaves Oc No cycles O d. More than one root

A: I have shown an example of rooted tree. The definition of rooted tree, tree might has leaves and it…

Q: Let T = (V, E) be a tree with V > 3 and k > 2 teaves. Prove that we can add k – 1 edges to T to…

A: Let TV, E be a tree with V≥3 and k≥2 leaves. Show that k-1 edges can be added to the graph of T to…

Q: 7. Prove: "If T is a tree of order n and size m, then m = n – 1." Give a concrete example to…

A:

Q: Suppose T is a binary tree of height H with n vertices. A. As a function of H, what are the minimum…

A: Solution for part a: Since T is a binary tree of height H with n vertices. In binary tree two cases…

Q: Prove that a tree with n,n≥2, vertices has at least two leaves.

A:

Q: An organization wants to assemble a cluster of n computers using wires and hubs. Each must have…

A: Solution of the problem as follows

Q: Explain why the following assertions, despite looking false, are actually true. (a) All graphs with…

A: (a). The given degree sequence of a graph is 6, 6, 5, 3, 2, 1, 1. We know that a graph is an…

Q: A 3-ary tree is a rooted tree where each parent has at most three children, and each child is…

A: In the given question we have to show that there is a a bijection between the set of non-isomorphic…

Q: Let Xn be a the number of graphs with vertex set [2n], Every vertex has a degree of 1. What is the…

A: For each Xn, we count graphs that satisfy two conditions: The graph must have a vertex set of 2n…

Q: 4. (a) (i) List all non-isomorphic trees (not rooted) on 6 vertices with no vertex of degree larger…

A: Consider the given information. (a) Now, draw all the non-isomorphic tress on 6 vertices with no…

Q: With reference to Exercise 1, suppose that there is athird step, and if the first step is made in…

A:

Q: Draw a Cayley graph for Z/7Z with generating set {2¯,5¯}.

A:

Q: There is no formula for the number of nonisomorphic trees on n vertices. But there is one for…

A: Given: There is no formula for the number of nonisomorphic trees on n vertices. But there is one for…

Q: A graph is called d-regular if every vertex has a degree of d. Let k and n be odd numbers, and prove…

A: Here we have to know the basic concept or the basic theorem before answering the question  THEOREM :…

Q: Who is the parent of vertex "6"in rooted tree (rooted at 1)? 2 3 4 None.

A:

Q: Consider the tree diagram. Path number 2 3 4 By C2 7. Find P(C182) - P(B2). (Enter your answer as a…

A: The tree diagram is given.

Q: 17. Danny studies for a test by taking notes of the text 10% of the time, highlighting the text 50 %…

A: Hello. Since your question has multiple sub-parts, we will solve the first three sub-parts for you.…

Q: 1 edges and hence use this to Prove by induction that a tree on n vertices has n – show that a…

A: