Q.1 Which of these are posets?
(a) (R; =)
(b) (R; <)
(c) (R; )
(d) (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
(a) Find the maximal elements.
(b) Find the minimal elements.
(c) Is there a greatest element?
(d) Is there a least element?
(e) Find all upper bounds of fa; b; cg.
1
(f) Find the least upper bound of fa; b; cg, if it exists.
(g) Find all lower bounds of ff; g; hg.
(h) Find the greatest lower bound of ff; g; hg, if it exists.
Q.3 Let G be a simple graph with n vertices.
(a) What is the maximum number of edges G can have?
(b) If G is connected, what is the minimum number of edges G can have?
(c) 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 Gn 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 = ;.
(a) Draw G5.
(b) Find the degree of each vertex in Gn.
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.
(a) Draw G0 for G = K4.
(b) 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
(ii) dist(u; v) = dist(v; u).
(iii) 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
(a) What is the sum of the entries in a row of the adjacency matrix for an undirected graph? For a directed graph?
(b) 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) K3;3 d) K3;4
3
