Graph. Notation: E = * *1, 2+, *1, 3+, *2, 3+, *3, 4+ +

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1 Graph graph is a pair (V, E) of two sets where V = set of elements called vertices (singl. vertex) E = set of pairs of vertices (elements of V) called edges Example: G = (V, E) where V = *,,, + E = * *, +, *, +, *, +, *, + + Notation: we write G = (V, E) for a graph with vertex set V and edge set E V(G) is the vertex set of G, and E(G) is the edge set of G drawing of G

2 Types of graphs Undirected graph E = **,+, *,+, *,+, *,++ SYMMETRIC & IRREFLEXIVE E(G) = set of unordered pairs Pseudograph (allows loops) loop = edge between vertex and itself Directed graph E(G) = set of ordered pairs Multigraph (IRREFLEXIVE) RELATION E(G) is a multiset E = **,+, *,+, *,+, E = **,+, *,+, *,+, *,+,,, *,+, *,+,,, *,+, *,+, *,+, *,++ *,++ 0 SYMMETRIC? E = *(,), (,), (,), (,), (,)+

3 More graph terminology simple graph (undirected, no loops, no parallel edges) for edge *u, v+ E(G) we say: u and v are adjacent u and v are neighbours u and v are endpoints of *u, v+ we write uv E G for simplicity N(v) = set of neighbours of v in G deg (v) = degree of v is the number of its neighbours, ie. N(v) is adjacent to and N() = *,+ N() = *,+ N() = *,,+ deg () = N() = *+ deg = deg () = deg () = = f() = f() = f() = G = (V, E ) is isomorphic to G = (V, E ) if there exists a bijective mapping f: V V such that uv E(G ) if and only if f u f(v) E(G )

4 Isomorphism 6 6 (,,,,,) degree sequence (,,,,,) (,,,,,) 6 if f is an isomorphism deg (u) = deg (f(u)) i.e., isomorphic graphs have same degree sequences only necessary not sufficient!!! (,,,,,,,) (,,,,,,,)

5 Isomorphism How many pairwise non-isomorphic graphs on n vertices are there? non-isomorphic graphs (without labels) all graphs (labeled) the complement of G = (V, E) is the graph G = (V, E) where E = u, v *u, v+ E+ E(G) = **,+, *,+, *,+, *,++ E(G ) = **,+, *,+, *,+, *,+, *,+, *,++ G G

6 Isomorphism How many pairwise non-isomorphic graphs on n vertices are there? self-complementary graph

7 Isomorphism Are there self complementary graphs on vertices?,,,... 6 vertices? No, because in a self-complementary graph G = (V, E) E = E and E + E = V 6 but = is odd 7 No, since =... 7 vertices?... 8 vertices? Yes, there are 0, 6,, 7, 6 8, 7, Yes,,,,

8 Handshake lemma Lemma. Let G = (V, E) be a graph with m edges. deg v = m v V Proof. Every edge *u, v+ connects vertices and contributes to exactly to deg (u) and exactly to deg (v). In other words, in v V deg (v) every edge is counted twice. How many edges has a graph with degree sequence: - (,,,,,)? - (,,,,)? - (0,,,)? Corollary. In any graph, the number of vertices of odd degree is even. Corollary. Every graph with at least vertices has vertices of the same degree. deg () = deg () = deg () = deg = m = 6

9 Handshake lemma Is it possible for a self-complementary graph with 00 vertices to have exactly one vertex of degree 0? Answer. No f isomorphism between G and G v is a unique vertex of degree 9 in G u the degree seq. of G and G (,,, 0,9,,, ) v G N(u) 0 V(G)\N(u) v, f(v) = u 9 we conclude f(u) = v N G (u) uv E(G) f(u)f(v) E(G ) but f(u)f(v) = vu E(G ) uv E(G) definition of isomorphism u G definition of complement 9

10 Subgraphs Let G = (V, E) be a graph a subgraph of G is a graph G = (V, E ) where V V and E E an induced subgraph (a subgraph induced on A V) is a graph G,A- = (A, E A ) where E A = E A A subgraph induced subgraph on A = *,,+

11 Walks, Paths and Cycles Let G = (V, E) be a graph a walk (of length k) in G is a sequence v, e, v, e,, v k, e k, v k where v,, v k V and e i = *v i, v i+ + E for all i *.. k + a path in G is a walk where v,, v k are distinct (does not go through the same vertex twice) a closed walk in G is a walk with v = v k (starts and ends in the same vertex) a cycle in G is a closed walk where k and v,, v k are distinct,{,},,{,},,{,}, is a walk (not path),{,},,{,},,{,}, is a cycle

12 Special graphs P n path on n vertices C n cycle on n vertices K n complete graph on n vertices B n hypercube of dimension n V B n = 0, n (a,, a n ) adjacent to b,, b n if a i s and b i s differ in exactly once n i= a i b i = (0,,0,0) adjacent to (0,,0,) not adjacent to (0,,,) P C K B

13 Connectivity a graph G is connected if for any two vertices u, v of G there exists a path (walk) in G starting in u and ending in v,{,},,{,}, path from to connected components no path from to connected not connected = disconnected a connected component of G is a maximal connected subgraph of G

14 Connectivity Show that the complement of a disconnected graph is connected! u u u v v w w G G G What is the maximum number of edges in a disconnected graph? k n k max k k + n k = n maximum when k = What is the minimum number of edges in a connected graph? path P n on n vertices has n edges cannot be less, why? keep removing edges so long as you keep the graph connected a (spanning) tree which has n edges

15 Distance, Diameter and Radius recall walk (path) is a sequence of vertices and edges v, e, v, e,, v k, e k, v k if edges are clear from context we write v, v,, v k length of a walk (path) is the number of edges distance d(u, v) from a vertex u to a vertex v is the length of a shortest path in G between u and v if no path exists define d u, v = d(u, v) d(,) = d(,) = matrix of distances in G distance matrix

16 Distance, Diameter and Radius eccentricity of a vertex u is the largest distance between u and any other vertex of G; we write ecc u = max d(u, v) v diameter of G is the largest distance between two vertices diam G = max d u, v = max ecc(v) u,v v radius of G is the minimum eccentricity of a vertex of G rad(g) = (as witnessed by called a centre) rad G = min v ecc(v) diam(g) = (as witnessed by, called a diametrical pair) Find radius and diameter of graphs P n, C n, K n, B n Prove that rad(g) diam(g) rad(g) ecc() = ecc() = ecc() = ecc() =

17 Independent set and Clique clique = set of pairwise adjacent vertices independent set = set of pairwise non-adjacent vertices clique number ω(g) = size of largest clique in G independence number α(g) = size of largest independent set in G *,,+ is a clique α G = ω G = *,+ is an independent set Find α(p n ), α(c n ), α K n, α(b n ) ω(p n ), ω(c n ), ω(k n ), ω(b n ) P

18 Bipartite graph a graph is bipartite if its vertex set V can be partitioned into two independent sets V, V ; i.e., V V = V bipartite not bipartite V V V V but now V is not an independent set because *,+ E(G) V

19 Bipartite graph a graph is bipartite if its vertex set V can be partitioned into two independent sets V, V ; i.e., V V = V K, K, K, Which of these graphs are bipartite: P n, C n, K n, B n? K n,m = complete bipartite graph V K n,m = *a a n, b b m + E K n,m = a i b j i.. n, j.. m + K,

20 Bipartite graph Theorem. A graph is bipartite it has no cycle of odd length. Proof. Let G = (V, E). We may assume that G is connected. any cycle in a bipartite graph is of even length 6 k odd k Assume G has no odd-length cycle. Fix a vertex u and put it in V. Then repeat as long as possible: u - take any vertex in V and put its neighbours in V, or - take any vertex in V and put its neighbours in V. Afterwards V V = V because G is connected. If one of V or V is not an independent set, then we find in G an odd-length closed walk cycle, impossible.? So, G is bipartite (as certified by the partition V V = V)

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