Written Assignment # 6 Solution

Description

Q.1 Which of these are posets?

1. (R; =)

2. (R; <)

3. (R; )

4. (R; 6=)

Q.2 Answer these questions for the partial order represented by this Hasse diagram.

l m

j	k
i	h	g
d	e	f
a	b	c

Figure 1: Q.2

1. Find the maximal elements.

2. Find the minimal elements.

3. Is there a greatest element?

4. Is there a least element?

5. Find all upper bounds of fa; b; cg.

6. Find the least upper bound of fa; b; cg, if it exists.

6. Find all lower bounds of ff; g; hg.

6. Find the greatest lower bound of ff; g; hg, if it exists.

Q.3 Let G be a simple graph with n vertices.

1. What is the maximum number of edges G can have?

2. If G is connected, what is the minimum number of edges G can have?

3. Show that if the minimum degree of any vertex of G is greater than or equal to (n 1)=2, then G must be connected.

Q.4 Let n 5 be an integer. Consider the graph G_n whose vertices are the sets fa; bg, where a; b 2 f1; : : : ; ng and a 6= b, and whose adjacency rule is disjointness, that is, fa; bg is adjacent to fa⁰ ; b⁰ g whenever fa; bg\fa⁰ ; b⁰ g = ;.

1. Draw G₅.

2. Find the degree of each vertex in G_n.

Q.5 Let G = (V; E)s be a graph on n vertices. Construct a new graph, G⁰ = (V ⁰ ; E⁰ ) as follows: the vertices of G⁰ are the edges of G (i.e., V ⁰ = E), and two distinct edges in G are adjacent in G⁰ if they share an endpoint.

1. Draw G⁰ for G = K₄.

2. Find a formula for the number of edges of G⁰ in terms of the vertex degrees of G.

Q.6 Let G be a connected graph, with the vertex set V . The distance between two vertices u and v, denoted by dist(u; v), is de ned as the minimal length of a path from u to v. Show that dist(u; v) is a metric, i.e., the following properties hold for any u; v; w 2 V :

(i) dist(u; v) 0 and dist(u; v) = 0 if and only if u = v.

2. dist(u; v) = dist(v; u).

2. dist(u; v) dist(u; w) + dist(w; v).

Q.7 Show that if G is bipartite simple graph with v vertices and e edges, then e v²=4.

Q.8

1. What is the sum of the entries in a row of the adjacency matrix for an undirected graph? For a directed graph?

2. What is the sum of the entries in a column of the adjacency matrix for an undirected graph? For a directed graph?

Q.9 Show that isomorphism of simple graphs is an equivalence relation.

Q.10 Show that every connected graph with n vertices has at least n 1 edges.

Q.11 Explain how to nd a path with the least number of edges between two vertices in an undirected graph by considering it as a shortest path problem in a weighted graph.

Q.12 Which of the these nonplanar graphs have the property that the removal of any vertex and all edges incident with that vertex produces a palnar graph? a) K₅ b) K₆ c) K_3;3 d) K_3;4

3