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notes-091203 - MATH 681 Notes Combinatorics and Graph...

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Unformatted text preview: MATH 681 Notes Combinatorics and Graph Theory I 1 Graph Theory 1.1 More on edge-counting The number of edges incident on a vertex will be relevant for a number of reasons. Definition 1. The degree of a vertex v in a graph G , denoted d G ( v ), is the number of edges in G which are incident upon v . The minimum degree in a graph G , denoted δ ( G ), is min v ∈ V ( G ) d G ( v ). The maximum degree , denoted Δ( G ), is max v ∈ V ( G ) d G ( v ). The subscripted G may be left out (and frequently is) if there is no ambiguity about which graph is being discussed. In older texts the degree may sometimes be called valence . Definition 2. If every vertex in a graph G has the same degree, G is called regular . A graph in which each vertex has degree k is specifically called k-regular . We can actually prove a cute theorem using graph degrees: Proposition 1. For any graph G , 2 k G k = ∑ v ∈ V ( G ) d ( v ) . Proof. 2 k G k = X e ∈ E ( G ) 2 2 k G k = X e ∈ E ( G ) X v ∈ V ( G ) 1 e incident on v 2 k G k = X v ∈ V ( G ) X e ∈ E ( G ) 1 e incident on v 2 k G k = X v ∈ V ( G ) |{ e ∈ E ( G ) : e incident on v }| 2 k G k = X v ∈ V ( G ) d ( v ) Note that what we wrote above applies inconsistently to nonsimple graphs; it would work if we “double-counted” loops in totalling up the degree. An interesting corollary: Corollary 1. If | G | and k are odd, G cannot be k-regular. Proof. If G were k-regular, then ∑ v ∈ V ( G ) d ( v ) = ∑ v ∈ V ( G ) k = | G | k would be odd. But 2 k G k must be even. Page 1 of 6 November 26, 2009 MATH 681 Notes Combinatorics and Graph Theory I 1.2 A Graph Library We’ve dispensed with the varied labelings of graphs, so we can talk about them in terms of structure only — that is to say, henceforth, our references to “graphs” will be descriptive of entire isomorphism classes. There are a couple of particularly notable ones: Definition 3. The complete graph K n is a graph on n vertices in which every pair of distinct vertices is in the edge set. In other words, the complete graph is the graph where every vertex is adjacent to every other vertex. It’s fairly easy to see that | K n | = n and k K n k = ( n 2 ) ; furthermore, K n is ( n- 1)-regular. Definition 4. The empty graph K c n is a graph on n vertices in which the edge-set is empty. We’ll see a justification for that superscripted “c” later. Clearly, | K c n | = n , k K c n k = 0, and K c n is 0-regular. Definition 5. The path P n is a graph on n vertices, denoted v 1 ,v 2 ,...,v n here for clarity, with the edge-set { [ v 1 ,v 2 ] , [ v 2 ,v 3 ] , [ v 3 ,v 4 ] ,..., [ v n- 1 ,v n ] } . Note the special case P 1 = K 1 = K c 1 . It is also the case that P 2 = K 2 . Furthermore, while identifying notable characteristics, it is worth mentioning that | P n | = n and k P n k = n- 1. As pertains to degree, δ ( P n ) = 1 and Δ( P n ) = 2....
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notes-091203 - MATH 681 Notes Combinatorics and Graph...

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