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MCA 520: Graph Theory Instructor Neelima Gupta [email protected] 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 • • • • • 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.