1. MCA 520: Graph Theory Instructor Neelima Gupta ngupta@cs.du.ac.in

2. Table of Contents Vertex Degrees and Counting

3. 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.

4. Degree-Sum Formula/First Theorem of Graph Theory/Handshaking Lemma Σ u 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)

5. K-dimensional cube or hypercube Q k Define the structure by way of k-tuple. Counting the number of vertices: ? Q k is ?-regular? Counting the number of edges:? Parity of a vertex: defined by the number of 1’s in its name Q k is bipartite…….Assignment

6. Q k continued Q k contains (k choose k-j) 2 k-j = (k choose j) 2 k-j subcubes isomorphic to Q j. Alternate argument for counting the number of edges in Q k : For j =1, Q 1 is nothing but an edge in Q k. The above formula thus gives us k 2 k-1 as the number of edges in Q k.

7. Recursive definition of Q k

8. 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 non- adjacent vertices have a common neighbour.

9. Bound is tight i.e there exists an example in which the min-deg < (n-1)/2 and the graph is disconnected. G = K floor(n/2) + K ceil(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.

10. Degree Sequence The degree sequence of a graph is the list of vertex degrees written in non-decreasing. Proposition: The non negative integers d 1 … d n are the vertex degrees of some graph iff Σd i 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.

11. 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.

12. 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 1- element graphic sequence is d 1 = 0.