Fundamentals of Graph Theory MATH Fundamentals of Graph Theory. Benjamin V.C. Collins, James A. Swenson MATH 2730

Transcription

1 MATH 2730 Fundamentals of Graph Theory Benjamin V.C. Collins James A. Swenson

2 The seven bridges of Königsberg Map: Merian-Erben [Public domain], via Wikimedia Commons

3 The seven bridges of Königsberg Map: Merian-Erben [Public domain], via Wikimedia Commons

4 Flatland utility lines Image: H. Dudeney, Amusements in Mathematics (1917)

5 Flatland utility lines Image: H. Dudeney, Amusements in Mathematics (1917)

6 Traveling Salesman Problem Image: Padberg-Rinaldi, 1987: 532 cities

7 What is not a graph? Example One of these two things is a graph in discrete mathematics: " %

8 What is a graph? Idea A graph is a bunch of points, some of which are joined to others by segments or arcs. Definition A graph is an ordered pair G = (V, E), such that V is a nonempty finite set, and E is a set of 2-element subsets of V. An element of V is a vertex, and an element of E is an edge. Example The figure above is a picture of G = ({r, s, t, u}, {{r, u}, {s, t}, {s, u}, {t, u}}).

9 What is a graph? Idea A graph is a bunch of points, some of which are joined to others by segments or arcs. Definition A graph is an ordered pair G = (V, E), such that V is a nonempty finite set, and E is a set of 2-element subsets of V. An element of V is a vertex, and an element of E is an edge. Consequences of the definition If e E, then e = {a, b} for some distinct a, b V. We say e is incident to a (and to b). We say a is an endpoint of e, but write a e. We say a is adjacent to b, and write a b. We also say a and b are neighbors.

10 What is a graph? Idea A graph is a bunch of points, some of which are joined to others by segments or arcs. Definition A graph is an ordered pair G = (V, E), such that V is a nonempty finite set, and E is a set of 2-element subsets of V. An element of V is a vertex, and an element of E is an edge. Consequences of the definition Every edge has two distinct endpoints (no loops). There is at most one edge between two given vertices.

11 Be careful with the pictures... These are two pictures of the same graph: G = ({r, s, t, u}, {{r, u}, {s, t}, {s, u}, {t, u}}).

12 How big is my graph? Definition Given a graph G = (V, E): The order of G is V (the number of vertices in G), occasionally called ν(g). The size of G is E (the number of edges in G), occasionally called ε(g). Definition Any graph of size 0 is called edgeless.

13 Each vertex has a degree Definition Let G = (V, E). The degree of a vertex is the number of neighbors it has. That is, if a V, then d(a) = {v V : v a}. Definition The set of neighbors of a V is called the neighborhood of a: N(a) = {v V : v a}, d(a) = N(a). Warning: a N(a).

14 Graph invariants Example a r s t u v w d(a) Definition Let G = (V, E) be a graph. The greatest vertex degree in G is denoted by (G); the least is δ(g).

15 Regular graphs Definition Let G = (V, E) be a graph. If all vertices have degree r, then (G) = r = δ(g) and we say G is r-regular. The Petersen graph

16 Complete graphs Theorem (and definitions) Suppose G = (V, E) is a graph of order n (so G has n vertices). The following are equivalent: 1. G is a complete graph on n vertices. 2. Any two distinct vertices of G are adjacent. 3. G is (n 1)-regular. 4. δ(g) = n E = ( n 2). 6. G = K n. K 5

17 Handshake theorem Theorem (The Handshake Theorem) If G = (V, E) is a graph, then d(v) = 2 E. v V Example a r s t u v w d(a)

18 Handshake theorem Theorem (The Handshake Theorem) If G = (V, E) is a graph, then d(v) = 2 E. v V Proof. We ask, How many half-edges does G contain? Clearly the answer is 2 E. On the other hand, for each v V, we see that d(v) is the number of half-edges incident on v. Since every half-edge is incident on a unique vertex, the total number of half-edges in G is d(v). v V