(a) Show that every tree on n vertices has exactly two 2-colorings.

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

A:

Q: How many pairwise non-isomorphic connected simple graphs are there on 6 vertices which contain two…

A: We have to find that how many pairwise non-isomorphic connected simple graphs are there on 6…

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: Explain why no such graph exists: (1): A graph with  degrees 1, 1, 2, 2, 3, 3, 5  (2): A connected…

A: To Prove - Explain why no such graph exists : A graph with degrees 1, 1, 2, 2, 3, 3, 5 A connected…

Q: 2. Show that any graph with at most 8 vertices is planar.

A: Topic:- planer graph

Q: Suppose T is a binary tree with 10° vertices. What are the minimum and maximum possible values of H,…

A: Given that T is a binary tree with 109 vertices. We have to find the minimum and maximum possible…

Q: Show that every graph with n vertices and k edges, n> k has n – k components.

A:

Q: a) Show that the number of leaves in an n-vertex tree with maximum degree A > 2 lies n(A–2)+2 A-1…

A: Given that a tree with n vertex with maximum degree ∆≥2. It is known that in tree every vertex is of…

Q: How many vertices and how many leaves does a full 5-ary tree with 100 internal vertices have?

A:

Q: 9. What is being described by "The smallest set of vertices, A, so that every vertex in the graph is…

A: Solution :-

Q: Suppose an n-vertex bipartite graph G has exactly k connected components, each of which has two or…

A: Given: The value of n vertex includes the bipartite graph G that has exactly k connected components…

Q: How many nonisomorphic (free) trees are there with 4 vertices?

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Show that there is a graph G with exactly 5 nodes where both G and its complement, have chromatic…

A:

Q: hich graph appears to contain all the orderad pairs in this rolation?

A:

Q: a) How many edges does a full 5-ary tree with 121 leaves have?

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 1. Draw pictures of all the isomorphism classes of trees with 5 vertices.

A:

Q: Let n be a positive integer. Show that every connected graph on n vertices has at least n – 1 -…

A: This is a problem of Graph Theory.

Q: If a graph G has n vertices, all of which but one have odd degree, how many vertices of odd degree…

A:

Q: 11. Show that a simple graph is a tree if and only if it is connected but the deletion of any of its…

A:

Q: Find a spanning tree for the connected graph. OA. OB. Oc. OD.

A: General properties of spanning tree: A spanning tree is a subset of Graph G, which has all the…

Q: Let G be a graph with n vertices. Prove that if G is connected and has n 1 edges, then G is a tree.…

A:

Q: 3. A labeled Ks graph with labeled edges appears below. Which edge permutation is induced by the…

A: Solution:

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

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: Suppose we add two edges in a tree T to form a graph G. How many cycles can be there in G.

A: When 1 edges are added to a tree,  a cycles are created When 2 edges are added to a tree, either 2…

Q: tree

A:

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: 2. Draw all non-isomorphic trees of order n, where 2 < n < 6.

A:

Q: Prove or disprove: There exists some n > 1 such that there is a simple graph on n vertices, where…

A:

Q: 2. Give an example of a directed graph with infinitely many vertices in which each verter has at…

A: A graph, i.e. a set of vertices or nodes that are connected together, where each edges are directed…

Q: A graph G with 7 vertices consists of two copies of K4 that share a single vertex. The vertices are…

A: The objective here is to find the number of distinct labeled spanning trees of G. Given: G is graph…

Q: QUESTION 4 A factory has two Machines-l and II. Machine-I produces 60% of items and Machine-Il…

A: Given that M=machine I, so machine II=McSo,…

Q: Prove that if a graph on n vertices is connected, and has n-1 edges, it must be acyclic.

A: A graph G is said to be a connected graph, if there is a path between any two vertices in that…

Q: Draw all forests of order 6.

A:

Q: Show that every simple graph on n vertices subgraph of K.

A:

Q: Show that every simple graph on n vertices is isomorphic to a subgraph of Kn.

A: We need to prove that every simple graph on n vertices is isomorphic to a subgraph of Kn (complete…

Q: a. Draw an acyclic graph with 7 vertices and 4 components. b. Prove that if G is an acyclic graph…

A: Given G is an acyclic graph with v vertices and 4 components.

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: 6. Find all nonisomorphic trees with five vertices.

A: All non-isomorphic trees with 5 vertices.

Q: A. If a simple graph G on 30 vertices has at least 301 edges, then its chromatic number is at least…

A: Note: As per bartleby instruction when more then one question is given only one has to be answered.…

Q: Let T be a tree of order n and suppose that all vertices of T have degree 1 or degree 3. Prove that…

A:

Q: G is a graph on 6 vertices with exactly

A: (b) G is a graph on 6 vertices with exactly two biconnected components:

Q: How many vertices and how many leaves does a complete m-ary tree of height h have?

A:

Q: Draw a graph with four vertices in which two vertices are of degree 2 and two vertices are of degree…

A: We have to solve given problem:

Q: 1. Consider the graph above. Give an ordering of the vertices so u apply the greedy algorithm to…

A: Given:

Q: How many possible graphs can be created on 5 distinct vertices?

A: Consider a graph of n vertices.The line connecting any two vertices of a graph is called as an edge…

Q: 2. Draw four spanning trees of graph G. d.

A:

Q: 2. Consider the trees with 7 vertices which have exactly 4 leaves. (a) Find a representative of each…

A:

Q: draw all non isomorphic trees with 4 vertices

A: A tree is a connected undirected graph. In a tree, any two vertices are connected by exactly one…

Q: How many non-isomorphic trees on 6 vertices are there? Draw a picture of

A: