notes-091126 - MATH 681 Notes Combinatorics and Graph...

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Unformatted text preview: MATH 681 Notes Combinatorics and Graph Theory I 1 Graph Theory On to the good stuff! Posets are but one kind of combinatorial structure. A somewhat more complicated structure is the graph , which describes relationships among a set of “nodes”. In applciation, graphs describe a myriad of different relationships: graphs have been used to explore physical connection (with nodes being cities, relationships being roads), logical connection (with nodes being webpages and relationships being links), or social connection (with nodes being people and relationships being acquaintance or collaboration). Mathematically, of course, we don’t much care what the relationships are, and treat them completely abstractly. Definition 1. A graph is an ordered pair ( V,E ), where V is a set of objects known as vertices or nodes and E is a set of unordered pairs of elements of V , which are called edges . The elements of V could be anything: numbers, named constants, even combinatorial objects in their own right! However, under most circumstances, we do not give the vertices highly distinctive names, and it’s conventional to refer to the elements of V as { v 1 ,v 2 ,...,v n } . Definition 2. If G = ( X,Y ), then the vertex-set X of graph G will be referred to as V ( G ); in cases where only one graph is under consideration, this may be shortened to simply V ; likewise the edge-set Y of graph G will be referred to as E ( G ), and simply as E when such a reference is unambiguous. Definition 3. A graph is finite if it has a finite number of vertices. The order of a finite graph is | V ( G ) | , also denoted | G | ; the number of edges | E ( G ) | in a finite graph G is denoted k G k . Unless otherwise mentioned, where the term “graph” is used, it denotes “finite graph”. 1.1 Edge-vocabulary Definition 4. Two vertices of a graph G are called adjacent (or adjacent in G , if the same vertex-set is used for multiple graphs) if [ x,y ] ∈ E ( G ). This may sometimes be denoted x ∼ y . Definition 5. An edge e of G is said to be incident upon a vertex v if e = [ v,x ] for some x . Definition 6. An edge e is said to be a loop if e = [ v,v ] for some vertex v . Definition 7. A graph G is simple if it has no loops. Unless otherwise mentioned, the term “graph” will henceforth denote simple graphs....
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This note was uploaded on 01/12/2012 for the course MATH 681 taught by Professor Wildstrom during the Fall '09 term at University of Louisville.

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notes-091126 - MATH 681 Notes Combinatorics and Graph...

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