1. Fundamental Graph Theory (Lecture 1) Lectured by Hung-Lin Fu 傅 恆 霖 Department of Applied Mathematics National Chiao Tung University

2. Lecture 1 : What is a graph? A graph G is an ordered pair (V,E) where V is a nonempty set of vertices (points or nodes) and E is a collection of multi-subsets (edges or lines) of V. If V is a finite set, then G = (V,E) is a finite graph and G is an infinite graph otherwise. If E is a collection of 2-element multi-subsets of V, then G is called a multi-graph. If E is a collection of 2-element subsets of V, then G is called a simple graph.

3. Continued … If E is a collection of 1-element (loop) or 2- element multi-subsets of V, then G is called a pseudo-graph. If E is a collection of multi-subsets of V, then G is known as a hypergraph. A hypergraph is said to be k-uniform if all subsets are of the same size k. Therefore a multi-graph (or a simple graph) is a 2-uniform hypergraph. We start our study with finite simple graph. (Why?)

4. Preliminaries Without mentioning otherwise, we let the graph we consider contain p vertices (order) and q edges (size). A (simple) graph has at most p(p-1)/2 edges, i.e. 0  q  p(p-1)/2. If q = 0, then we have an empty graph and if q = p(p-1)/2, then we have a complete graph, denoted by O p and K p respectively.

5. An example p = 10 and q = 9.

6. Neighborhood and Degree Let v be a vertex of G = (V,E). The neighborhood of v in G, denoted by N G (v), is the set of vertices u in G such that {u,v} (uv in short) is an edge of G, i.e. N G (v) = {u  V: uv  E}. For convenience, we use V(G) and E(G) to denote the vertex set and edge set of G respectively. If uv is an edge of G, then we say u and v are adjacent or u is incident to uv, so is v. The degree of v in G, denoted by deg G (v), is the cardinality of N G (v), i.e. deg G (v) = | N G (v) |.

7. The 1 st theorem of Graph Theory Theorem 1.1  v  V(G) deg G (v) = 2|E(G)| = 2q. Proof. Every edge provides 2 degrees.  The sum of degrees in a graph G is also defined as the volume of G, denoted by Vol(G), i.e. Vol(G) = 2q. Corollary 1.2 The number of vertices with odd degree is even. Corollary 1.3 There does not exist a simple graph in which all vertices are of distinct degrees. (*) How about a multi-graph?

8. Odd and Even graphs A graph is an odd (respectively even) graph if each vertex of the graph is of odd (respectively even) degree. A graph is an r-regular graph if each vertex is of degree r. A 3-regular graph is also known as a cubic graph. No odd graph is of odd order! (?)

9. Subgraphs A graph H = (V´,E´) is a subgraph of G = (V,E) provided that V´  V and E´  E. We say a graph G is defined on V if G = (V,E). A subgraph H of G is an induced subgraph of G if two vertices of V(H) are adjacent in G, then they are adjacent in H. H is also said to be generated by S = V(H) in G, denoted by G. Give an example of subgraph which is not an induced subgraph.

10. Beautiful Graphs Let V = Z n and two vertices i and j are adjacent if and only if |i – j|  2 (mod n). Then we have a graph of size n which is defined on Z n. (?) The graph obtained above is known as a graph defined on Z n with difference 2. Of course, it is a subgraph of the complete graph defined on Z n. The above graph is denoted by Z n [{2}]. Can you define Z n [S] for a subset S of Z n ?

11. More Beautiful Graphs Let V(G) be the set of all 3-subsets of Z n and two vertices in G are adjacent if and only if they are mutually disjoint. Then G is of order p = C(n,3) and of size q = p C(n-3,3)/2. ( C(x,y) denotes the number of ways of choosing y objects from x objects.) If n = 2k+1 and V(G) is the set of all k-subsets of Z n, then the graph defined as above is known as the Johnson graph (Odd graph) of order 2k+1.

12. Generalized Johnson Graphs Let G be the graph with vertex-set the set of all k-subsets of Z n and two vertices are adjacent if and only if their intersection has cardinality h (  0). We denote the above graph by GJ(n,k,h). Therefore, the Johnson graph of order 2k+1 is a GJ(2k+1,k,0). The graph defined above is the so-called intersection graph. Example: Find GJ(7,3,1).

13. What a beautiful graph it is!

14. Supergraphs If H is a subgraph of G, then G is called a supergraph of H. The maximum and minimum degree of a graph G is denoted by  (G) and  (G) respectively. The deficiency def(G) =  v  V(G) (  (G) – deg G (v)). Problem: Let H be a simple graph. Find a way to obtain a simple supergraph G of H which is  (H)-regular with minimum size and H is an induced subgraph of G. Note: If H is a regular graph, then we do nothing.

15. Directed Graphs A directed graph D defined on V is an ordered pair (V,A) where V is the vertex set and A  V×V is the arc set. If V has p elements (vertices), then A has at most p 2 elements (arcs). If (u,v) is an arc of D, then we use the following figure to depict the arc. u v

16. Complete Directed Graphs The complete directed graph (digraph) of order p is the digraph (V,A) where V is a p-set and A contains all ordered pairs in V×V, denoted by D p. In A, (v,v) is a directed loop from v into v and we say that u is adjacent from u into v if (u,v) is an arc of A. A digraph (V,A) is a tournament if for each pair of distinct elements u and v in V exactly one of the two arcs (u,v) and (v,u) is in A, denoted by T p.

17. In-degree and Out-degree Let G = (V,A) be a digraph. Then the in- degree of a vertex v, deg – (v) is the number of u ’ s where (u,v) is in A and the out-degree of v, deg + (v) is the number of w ’ s where (v,w) is in A. If there is no confusion we omit the sub-index G for the degree. Theorem 1.4 In a digraph G,  v  V deg – (v) =  v  V deg + (v) = |A| = |A(G)|. An s-regular digraph G is a digraph in which for each vertex v, deg + (v) = deg – (v). Thus, an s-regular digraph of order p has p s arcs.

18. An Important Example Let G = (V,A) be a digraph defined on Z n, i.e. V = Z n and the element i is adjacent to j if and only if j  i (mod n) is in S where S is a subset of Z n *. Then G has s n arcs provided that S is a set with s elements. (Z n * = Z n \ {0}.) This is a well-known network. If s = 2, then we have a double-loop network.

19. Weighted Graphs A weighted graph G is a pair (G,  ) where G is a graph (or a digraph) and  is a function from the edge set (or arc set) of G into R. (R denotes the set of real numbers.) A network is a weighted digraph with two designated vertices of the vertex set called source and sink respectively.

20. Signed Graphs A signed graph is a pair (G,  ) where G is a simple graph and  is a function from E(G) into {+,  }. A signed graph is very useful in modeling mathematical problems raised from social sciences, say political sciences and sociology; “ + ” represents a good connection (for example, in the same party or race) and “  ” otherwise.

21. Labeled Graphs Without mention otherwise, a graph is considered as a labeled graph with each vertex labeled. Two graphs G and G are isomorphic if there exists a bijection  from V(G) onto V(G) such that uv  E(G)   (u)  (v)  E(G). If two graphs are isomorphic, then there is a way of drawing the two graphs exactly the same.

22. Isomorphism To determine whether two graphs are isomorphic is not easy at all. (It is known as the isomorphism disease.) Problem. How many non-isomorphic graphs are there of order 5 and size 7? Let G  G denote that G and G are isomorphic and  denotes the set of all graphs. Then  is an equivalence relation in .

23. General Subgraphs A graph H is a subgraph of a graph G if H is isomorphic to a subgraph of G. With this definition, a graph without labels can be recognized. In other words, without mention for special purpose, all graphs we consider are graphs without labels, or equivalently, we can arbitrarily give labels to the set of vertices of a graph. Note that all labels of the vertices are distinct.

24. Graph Representations Adjacency Matrix A(G) Let the vertex set of G be {v 1,v 2, …, v p }. Then A(G) = [a i,j ] pxp where a i,j = 1 if v i is adjacent to v j in G and a i,j = 0 otherwise. If G is a simple graph, then A(G) is a symmetric matrix with all 0 ’ s diagonal. On the other hand, if G is a digraph, then A(G) may not be symmetric and also some entries on the diagonal are not 0 ’ s. Note: The multi-set of eigenvalues of A(G) is defined to be the spectrum of G, dented by Spec(G).

25. Continued … Incidence Matrix B(G) Let the vertex-set of G be {v 1,v 2, …, v p } and the edge-set (or arc set) be {e 1,e 2, …, e q }. Then B(G) = [b i,j ] pxq where b i,j = 1 if v i is incident to e j, and b i,j = 0 otherwise. If G is a simple graph, then each column of B(G) has exactly two 1 ’ s. The row sum of B(G) is a zero vector provided the addition is taken modulo 2. If G is a hypergraph, then the above property may not hold.

26. Walks in Graphs A walk from u to v in a graph G is a sequence of vertices in G where v i v i+1 is an edge of G for i = 0,1,2, …,t – 1. A trail is a walk such that all edges v i v i+1 are distinct and a path is a trail in which all vertices are distinct. A closed trail (u = v) is called a circuit and a circuit without repeating a vertex is a cycle. The length of a walk (or a trail, a circuit, a path, a cycle) is the number of edges in the walk (or respectively trail, … ).

27. The Number of Walks Theorem 1.5 Let G be a graph and V(G) = {v 1,v 2, …, v p }. Then the number of walks of length n (  1) from v i to v j is the (i,j)- entry of [A(G)] n. Proof. By induction on n. Problem Find the number of cycles of length 3 and 4.

28. Distance in Graphs The distance of two vertices u and v in G, denoed by d G (u,v), is the length of a shortest path from u to v, i.e. a path with minimum number of edges. The eccentricity of a vertex v in a graph G, e G (v) = max{d G (u,v)| u  V(G)}. The diameter of G, diam(G) = max{e G (v) | v  V(G)}. The radius of G, rad(G) = max{e G (v) | v  V(G)}. Theorem 1.6 rad(G)  diam(G)  2rad(G).

29. Concluding Remark of Lecture 1 Graph Theory has developed into a very active area of mathematical research. Almost everyday, a new idea of applying graph theory to solve a problem in real world will be discovered. It is up to you in the near future to find yet another one or even more. This department is one of the best places in the world that you can learn this topic!