Discrete Mathematics (Math 510) — Fall 2014 Definitions and Theorems Gerald Hoehn October 12, 2014 Chapter 1 Graph Theory 1.1 Basics Definition 1.1. A graph G is an ordered pair (V,E) of disjoint finite sets such that E is a subset of the set of unordered pairs of V . We call V = V (G) the vertices of G and E = E(G) the edges of G. An edge {x, y} in E is said to join the vertices x and y and denoted by xy. We call x and y the end-vertices of the edge xy. Two vertices x and y are called adjacent if xy is an edge of G. The order of G is the cardinality of V , the size of G is the cardinality of E. Definition 1.2. A complete graph is a graph in which all distinct vertices are adjacent. The complete graph of order n is denoted by Kn. Definition 1.3. A empty graph is a graph without edges. The empty graph of order n is denoted by En. Definition 1.4. The complement G of a graph G is the graph with the same vertex set as G, and whose edge set consists of all edges not present in G. Definition 1.5. Cycle on n vertices, Cn; defined by picture (vertices v1, ..., vn, edges v1v2, v2v3, ..., vn−1vn, vnv1). Definition 1.6. Path on n-vertices, Pn; defined by picture (vertices v1, ..., vn, edges v1v2, v2v3, ..., vn−1vn). Definition 1.7. A graph G is bipartite if its vertex set can be partitioned into two sets X and Y such that every edge of G has one end vertex in X and the other in Y . The complete bipartite graph K|X|,|Y | is the graph with vertex set V = · X S Y and edge set E = {xy | x ∈ X, y ∈ Y }. Definition 1.8. Two graphs G and H are said to be isomorphic if there exists a one-to-one correspondence f : V (G) −→ V (H) such for each pair of vertices x, y ∈ V (G) one has xy ∈ E(G) if and only if f(x)f(y) ∈ E(H). One calls f an isomorphism. 1 “Isomorphic” is a precise way to say that we “ignore” the labels of the vertices. Definition 1.9. We say that a graph H is a subgraph of a graph G if V (H) ⊂ V (G) and E(H) ⊂ E(G). We also say that H is a subgraph if H is isomorphic to a subgraph in the above sense. Definition 1.10. For G a graph and v a vertex the degree deg(v) is the number of edges with end-vertex v. We let δ(G) be the minimal and ∆(G) be the maximal degree of any vertex of G. Theorem 1.1. The sum of the degrees of the vertices is equal to twice the numbers of edges: deg(v) = 2 ·|E(G)|. v∈V (G) Definition 1.11. A walk in a graph is a sequence of vertices v1, v2, ..., vk such that vivi+1 is an edge for each i = 1, ..., k − 1. A path is a walk in which all vertices are distinct. A cycle is path v1, ..., vk where k ≥ 3 and vkv1 is also an edge. The length of a walk, path or cycle is the number of corresponding edges. Theorem 1.2. If in a graph there exists a walk from u to v, then there exists also a path from u to v. Definition 1.12. Let G be a graph. For an edge e of G, G − e is the graph with the same vertices and all edges besides e. For an vertex v of G, G − v is the graph obtained by removing v and all edges ending in v. Definition 1.13. A graph is called connected if every two vertices can be joined by a path. Otherwise it is called disconnected. Note that every graph is the disjoint union of connected components. Definition 1.14. The connectivity κ(G) of a graph G is the minimal num- ber of vertices v1, ..., vk one has to choose such that G − {v1,...,vk} is disconnected. A graph is called k-connected if k ≤ κ(G). Note that a graph is connected if and only if κ(G) ≥ 1, i.e., G is 1- connected. Theorem 1.3. A graph is bipartite if and only if it contains no cycle of odd length. 2 1.2 Distance General assumption in this section: G is connected. Definition 1.15. Let u, v be two vertices of a graph G. The distance between u and v is the length of the shortest path connecting u with v and denoted by dG(u,v) or just d(u,v). This implies that d(u,u) = 0 for any vertex u. Definition 1.16. The diameter of a graph G, denoted diam(G), is the greatest distance between any two vertices: diam(G)= max d(u,v). u,v∈V (G) Theorem 1.4. On has: (i) d(u,v) ≥ 0 for all u, v and d(u,v) = 0 if and only if u = v. (ii) d(u,v)= d(v,u) for all u, v. (iii) d(u,w) ≤ d(u,v)+ d(v,w) for all u, v, w. Theorem 1.5. Let u be any vertex in a graph G. Let m be the maximal distance between u and any other vertex. Then m ≤ diam(G) ≤ 2m. Definition 1.17. Let G be a graph with vertices v1, ..., vn. The adjacency matrix of G is the n × n matrix whose (i,j) entry, denoted by [A]i,j, is defined by 1, if vi and vj are adjacent, [A]i,j = 0, otherwise. For graphs (without loops) the entries on the diagonal of A are zero. The matrix is symmetric, i.e., [A]i,j =[A]j,i for all i and j. Definition 1.18. Let A and B two n × n matrices. The product matrix A · B is the n × n matrix with entries [A · B]i,j =[A]i,1 · [B]1,j +[A]i,2 · [B]2,j + ··· +[A]i,n · [B]n,j. We set shortly A2 = A · A and Ak = Ak−1 · A for k ≥ 2. Theorem 1.6. Let G be a graph with vertices v1, v2, ..., vn and correspond- ing adjacency matrix A. Then for any positive integer k, the (i,j) entry of k A equals the number of walks from vi to vj that use exactly k edges. Corollary: Let G be a graph with vertices v1, v2, ..., vn and corresponding k adjacency matrix A. If d(vi,vj)= m, then [A ]i,j = 0 for 1 ≤ k < m. 3 Definition 1.19. For an n × n matrix A and a positive integer k set 2 k Sk = I + A + A + ··· + A where I is the identity matrix: 1 0 ... 0 0 0 1 ... 0 0 . I = . 0 0 ... 1 0 0 0 ... 0 1 . Theorem 1.7. Let G be a graph with vertices v1, ..., vn. If m is the smallest positive integer such that all entries of Sm are positive, then diam(G)= m. 1.3 Trees Definition 1.20. • A forest is a graph containing no cycles as subgraphs. • A tree is a connected forest. • A leave is a vertex of degree 1 in a tree. Theorem 1.8. A tree T of order n has n − 1 edges. Theorem 1.9. A graph of order n is a tree if and only if it is connected and contains n − 1 edges. Theorem 1.10. A graph of order n is a tree if and only if it contains no cycles and contains n − 1 edges. Theorem 1.11 (Cayley’s Tree formula). There are nn−2 distinct labeled trees of order n. Definition 1.21. A Pruefer seqence of a labeled tree is a sequence a1, a2, ..., an−2 of numbers from {1, 2,...,n} assigned in a unique way to a tree. (See the book for the precise algorithm.) A labeled tree can be uniquely reconstructed from its Pruefer sequence. (See the book for the precise algorithm.) Definition 1.22. The degree matrix of a graph G is the n × n matrix D defined by deg(vi), if i = j, [D]i,j = 0, else. Definition 1.23. Given an n × n-matrix M, the (i,j)-cofactor of M is defined to be (−1)i+jdet(M(i|j)) where det(M(i|j)) is the determinant of the (n − 1) × (n − 1)-matrix one gets by deleting the i-th row and the j-th column from M. 4 Theorem 1.12 (Matrix Tree Theorem (Kirchhoff)). Let G be a connected labeled graph with adjacency matrix A and degree matrix D. Then the num- ber of unique spanning trees of G is equal to the value of any cofactor of the matrix D − A. 1.4 Eulerian and Hamiltonian walks Eulerian trails and circuits Definition 1.24. Let G be a graph. A trail in G is a walk that does not repeat any edges. A closed trail is called a circuit. A cycle is a circuit. Definition 1.25. An Eulerian trail (resp. Eulerian circuit) in a graph G is a trail (resp. circuit) in G which includes every edge. A graph that contains an Eulerian circuit is said to be an Eulerian graph. Theorem 1.13. For a connected graph G the following statements are equiv- alent: (1) G is Eulerian.