MCA 520: Graph Theory
Instructor
Neelima Gupta
ngupta@cs.du.ac.in
Table of Contents
Vertex Degrees and Counting
Regular Graphs
• A graph is regular iff deg(u) = deg(v) for all
vertices u and v.
• Alternate definition: A graph is regular iff
min_deg = max_deg.
• A graph is k-regular iff deg(u) = k for all
vertices u.
Degree-Sum Formula/First Theorem of
Graph Theory/Handshaking Lemma
• Σu in V(G) deg(u) = 2 |E|
• In a graph G, average vertex degree = 2 |E|/|V|.
• Min-deg(G) < 2 |E|/|V| < max-deg(G)
K-dimensional cube or hypercube Qk
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Define the structure by way of k-tuple.
Counting the number of vertices: ?
Qk is ?-regular?
Counting the number of edges:?
Parity of a vertex: defined by the number of
1’s in its name
• Qk is bipartite…….Assignment
Qk continued
• Qk contains (k choose k-j) 2k-j = (k choose j) 2k-j
subcubes isomorphic to Q j.
• Alternate argument for counting the number
of edges in Qk : For j =1, Q1 is nothing but an
edge in Qk. The above formula thus gives us k
2k-1 as the number of edges in Qk.
Recursive definition of Qk
Extremal Problems
• The minimum number of edges in a connected
graph with n vertices is n -1.
• If G is a simple n vertex graph with min-deg(G)
> (n-1)/2, then G is connected.
• Proof : We’ll show that every pair of nonadjacent vertices have a common neighbour.
Bound is tight
• i.e there exists an example in which the min-deg < (n-1)/2
and the graph is disconnected.
• G = Kfloor(n/2) + Kceil(n/2)
• min-deg(G) = floor(n/2) -1
• And G is disconnected.
• Thus minimum value of min-deg(G) that forces a simple
graph to be connected is floor(n/2)
• Or
• The maximum value of min-deg(G) in a disconnected
simple graph is floor(n/2) -1.
Degree Sequence
• The degree sequence of a graph is the list of
vertex degrees written in non-decreasing.
• Proposition: The non negative integers d1 … dn
are the vertex degrees of some graph iff Σdi is
even.
• Sufficiency is true if loops are allowed. In a
simple graph (loops are not allowed) 2,0,0 is
not realizable, though sum of degrees is even.
Graphic Sequence
• GS is a DS that is realizable by a simple graph.
• Example 1:
– 1 0 1 is graphic
– 2211 is graphic
• Example 2: Test whether 33333221 is graphic.
– Reduce 33333221 to 2223221 rearranged as 3222221
– Reduce 3222221 to 111221 rearranged as 221111
– Reduce 221111 to 10111 rearranged as 11110. It is
easy to show that this is realizable.
Theorem : Havel and Hakimi
• For n > 1, an integer list d of size n is graphic iff
d’ is graphic, where d’ is obtained from d by
deleting its largest element Δ and subtracting
1 from its next largest Δ elements. The only 1element graphic sequence is d1 = 0.