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LKs\ާE=kL R6V%HPx …,׶LM7`U&LQo"`H9پـzawq1v&( wwwwnF8!ܢsilPNG  IHDR exgIFxNETSCAPE2.0$NPPLTE3f333f3333f3ffffff3f̙3f3f333f333333333f33333333f33f3ff3f3f3f3333f33̙33333f3333333f3333f3ffffff3f33ff3f3f3f3fff3ffffffffffff3ffff̙fff3fffffff3ffffff3f333f3333f3ffffff3f̙̙3̙f̙̙̙̙3f3f̙333f3̙333f3fff̙fff3f̙̙3f̙3f̙3f333f3333f3ffffff3f̙3f3f9ųtRNS[rbKGDHgIFgdni cmPPJCmp0712HsyIDATWMϱ0?:JU9bW&":"ӯxH}fzIcv?@~9ٛ"}3S lvE\ĽU!b$CV`@!d,3f333f3333f3ffffff3f̙3f3f333f333333333f33333333f33f3ff3f3f3f3333f33̙33333f3333333f3333f3ffffff3f33ff3f3f3f3fff3ffffffffffff3ffff̙fff3fffffff3ffffff3f333f3333f3ffffff3f̙̙3̙f̙̙̙̙3f3f̙333f3̙333f3fff̙fff3f̙̙3f̙3f̙3f333f3333f3ffffff3f̙3f3f *@X JÇ8 E/^"BdQO^ b3X"(Ó CfL37~eE  | 8 PreliminariesZWithout 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 Op and Kp respectively.J.B 9$ %    An example p = 10 and q = 9.Neighborhood and DegreeLet v be a vertex of G = (V,E). The neighborhood of v in G, denoted by NG(v), is the set of vertices u in G such that {u,v} (uv in short) is an edge of G, i.e. NG(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 degG(v), is the cardinality of NG(v), i.e. degG(v) = | NG(v) |.ZHU  'j  %   &&&t}1kF-' The 1st theorem of Graph Theory* Theorem 1.1 Sv V(G) degG(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? Z ,, 07 / 4 F  Odd and Even graphsA 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! (?) ZQE  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 <S>G. Give an example of subgraph which is not an induced subgraph.v""&""",@b _    Beautiful Graphs Let V = Zn 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 Zn. (?) The graph obtained above is known as a graph defined on Zn with difference 2. Of course, it is a subgraph of the complete graph defined on Zn. The above graph is denoted by Zn[{2}]. Can you define Zn[S] for a subset S of Zn? yZZ Z; C* {  More Beautiful Graphs"Let V(G) be the set of all 3-subsets of Zn 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 Zn, then the graph defined as above is known as the Johnson graph (Odd graph) of order 2k+1.Z) "&x"*2"&"" Generalized Johnson GraphsLet G be the graph with vertex-set the set of all k-subsets of Zn 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).|Y@ S$ j   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 D(G) and d(G) respectively. The deficiency def(G) = S vV(G) (D(G)  degG (v)). Problem: Let H be a simple graph. Find a way to obtain a simple supergraph G of H which is D(H)-regular with minimum size and H is an induced subgraph of G. Note: If H is a regular graph, then we do nothing.zZ     U     :b  F D 9Directed GraphsA directed graph D defined on V is an ordered pair (V,A) where V is the vertex set and A VV is the arc set. If V has p elements (vertices), then A has at most p2 elements (arcs). If (u,v) is an arc of D, then we use the following figure to depict the arc.tI 6_3(Complete Directed GraphsThe 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 VV, denoted by Dp. 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 Tp.ZZ t,4)In-degree and Out-degreeLet 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, S vVdeg (v) = S vVdeg+(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. ZB*I /  Z4 Z Z * 7 *1" 4 5>An Important ExamplenLet G = (V,A) be a digraph defined on Zn, i.e. V = Zn 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 Zn*. Then G has s" n arcs provided that S is a set with s elements. (Zn* = Zn \ {0}.) This is a well-known network. If s = 2, then we have a double-loop network.(8& 4*2 $   $8 $ J?Weighted Graphs<A weighted graph G is a pair (G,w) where G is a graph (or a digraph) and w 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.$9$MK@ Signed GraphsA signed graph is a pair (G,e) where G is a simple graph and e 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. XDILALabeled GraphstWithout 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 j from V(G) onto V(G) such that uv E(G) j(u)j(v) E(G). If two graphs are isomorphic, then there is a way of drawing the two graphs exactly the same. ;U 0` , !zMB Isomorphism jTo 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 W denotes the set of all graphs. Then @ is an equivalence relation in W. 6#JTPEGeneral Subgraphs zA 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.J{L %, % 5NCGraph RepresentationsAdjacency Matrix A(G) Let the vertex set of G be {v1,v2, & , vp}. Then A(G) = [ai,j]pxp where ai,j = 1 if vi is adjacent to vj in G and ai,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).,ZZ           c<  ?ODContinued &  6Incidence Matrix B(G) Let the vertex-set of G be {v1,v2, & , vp} and the edge-set (or arc set) be {e1,e2, & , eq}. Then B(G) = [bi,j]pxq where bi,j = 1 if vi is incident to ej, and bi,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. FZ   %          UO>j;.& )RGWalks in GraphsA walk from u to v in a graph G is a sequence <u = v0, v1, v2, & , vt = v> of vertices in G where vivi+1 is an edge of G for i = 0,1,2, & ,t  1. A trail is a walk such that all edges vivi+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, & ). Z!      $"   3 /t>BP)SHThe Number of Walks Theorem 1.5 Let G be a graph and V(G) = {v1,v2, & , vp}. Then the number of walks of length n ( 1) from vi to vj 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.V)33 ,33,49uTIDistance in GraphsThe distance of two vertices u and v in G, denoed by dG(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, eG(v) = max{dG(u,v)| u V(G)}. The diameter of G, diam(G) = max{eG(v) | v V(G)}. The radius of G, rad(G) = max{eG(v) | v V(G)}. Theorem 1.6 rad(G) diam(G) 2rad(G). $pZ)^ &3333333333&33 333333333333&333333  Z Z33333333 Z+ % " QFConcluding Remark of Lecture 1 VGraph 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. 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MMM̙f  ,6(  ,x , c $?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSVWRoot EntrydO)"CUPicturesICurrent User,SummaryInformation(LRPowerPoint Document(3EDocumentSummaryInformation8rowan\1|d0|Wo 0`DjwiԚolarrowan\1|d0|Wo 0A ` .  @n?" dd@  @@`` t\ $  @ "$!$&'($$$R$AQII([VMt$$$r$wq1v&itb$8!ܢsilF2u 0P33 Z @812j ]oʚ;2Nʚ;g4BdBd0p ppp@ <4!d!d 0,H2<4dddd 0,H2 +#___PPT9nMZ4&3SwкPNG  IHDRdPLTEϚe3fhhΙFtRNSS cmPPJCmp0712TIDATm 1 J78@h!tQyDAP9rORHFz  | 8 PreliminariesZWithout mentioning otherwise, we let the graph we consider contain p vertices (order) and q edges (size). A (simple) graph has at most p(p-1)/2 edges, i.e. 0 q p(p-1)/2. If q = 0, then we have an empty graph and if q = p(p-1)/2, then we have a complete graph, denoted by Op and Kp respectively.J.B 9$ %    An example p = 10 and q = 9.Neighborhood and DegreeLet v be a vertex of G = (V,E). The neighborhood of v in G, denoted by NG(v), is the set of vertices u in G such that {u,v} (uv in short) is an edge of G, i.e. NG(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 degG(v), is the cardinality of NG(v), i.e. degG(v) = | NG(v) |.ZHU  'j  %   &&&t}1kF-' The 1st theorem of Graph Theory* Theorem 1.1 Sv V(G) degG(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? Z ,, 07 / 4 F  Odd and Even graphsA 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! (?) ZQE  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 <S>G. Give an example of subgraph which is not an induced subgraph.v""&""",@b _    Beautiful Graphs Let V = Zn 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 Zn. (?) The graph obtained above is known as a graph defined on Zn with difference 2. Of course, it is a subgraph of the complete graph defined on Zn. The above graph is denoted by Zn[{2}]. Can you define Zn[S] for a subset S of Zn? yZZ Z; C* {  More Beautiful Graphs"Let V(G) be the set of all 3-subsets of Zn 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 Zn, then the graph defined as above is known as the Johnson graph (Odd graph) of order 2k+1.Z) "&x"*2"&"" Generalized Johnson GraphsLet G be the graph with vertex-set the set of all k-subsets of Zn 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).|Y@ S$ j   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 D(G) and d(G) respectively. The deficiency def(G) = S vV(G) (D(G)  degG (v)). Problem: Let H be a simple graph. Find a way to obtain a simple supergraph G of H which is D(H)-regular with minimum size and H is an induced subgraph of G. Note: If H is a regular graph, then we do nothing.zZ     U     :b  F D 9Directed GraphsA directed graph D defined on V is an ordered pair (V,A) where V is the vertex set and A VV is the arc set. If V has p elements (vertices), then A has at most p2 elements (arcs). If (u,v) is an arc of D, then we use the following figure to depict the arc.tI 6_3(Complete Directed GraphsThe 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 VV, denoted by Dp. 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 Tp.ZZ t,4)In-degree and Out-degreeLet 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, S vVdeg (v) = S vVdeg+(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. ZB*I /  Z4 Z Z * 7 *1" 4 5>An Important ExamplenLet G = (V,A) be a digraph defined on Zn, i.e. V = Zn 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 Zn*. Then G has s" n arcs provided that S is a set with s elements. (Zn* = Zn \ {0}.) This is a well-known network. If s = 2, then we have a double-loop network.(8& 4*2 $   $8 $ J?Weighted Graphs<A weighted graph G is a pair (G,w) where G is a graph (or a digraph) and w 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.$9$MK@ Signed GraphsA signed graph is a pair (G,e) where G is a simple graph and e 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. XDILALabeled GraphstWithout 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 j from V(G) onto V(G) such that uv E(G) j(u)j(v) E(G). If two graphs are isomorphic, then there is a way of drawing the two graphs exactly the same. ;U 0` , !zMB Isomorphism jTo 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 W denotes the set of all graphs. Then @ is an equivalence relation in W. 6#JTPEGeneral Subgraphs zA 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.J{L %, % 5NCGraph RepresentationsAdjacency Matrix A(G) Let the vertex set of G be {v1,v2, & , vp}. Then A(G) = [ai,j]pxp where ai,j = 1 if vi is adjacent to vj in G and ai,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).,ZZ           c<  ?ODContinued &  6Incidence Matrix B(G) Let the vertex-set of G be {v1,v2, & , vp} and the edge-set (or arc set) be {e1,e2, & , eq}. Then B(G) = [bi,j]pxq where bi,j = 1 if vi is incident to ej, and bi,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. FZ   %          UO>j;.& )RGWalks in GraphsA walk from u to v in a graph G is a sequence <u = v0, v1, v2, & , vt = v> of vertices in G where vivi+1 is an edge of G for i = 0,1,2, & ,t  1. A trail is a walk such that all edges vivi+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, & ). Z!      $"   3 /t>BP)SHThe Number of Walks Theorem 1.5 Let G be a graph and V(G) = {v1,v2, & , vp}. Then the number of walks of length n ( 1) from vi to vj 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.V)33 ,33,49uTIDistance in GraphsThe distance of two vertices u and v in G, denoed by dG(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, eG(v) = max{dG(u,v)| u V(G)}. The diameter of G, diam(G) = max{eG(v) | v V(G)}. The radius of G, rad(G) = max{eG(v) | v V(G)}. Theorem 1.6 rad(G) diam(G) 2rad(G). $pZ)^ &3333333333&33 333333333333&333333  Z Z33333333 Z+ % " QFConcluding Remark of Lecture 1 VGraph 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! @WZLjJ33W 0$0(  $r $ S   ~ $ s *\  0  H $ 0޽h ? MMM̙f  l$(  lr l S |.;P   r l S $/<  0  H l 0޽h ? @AvffJff  `$(  `r ` S A; P   r ` S B<  H ` 0޽h ? @AvffJffr  ?QBTADT ՜.+,0    0 pùjpA-Mathj3E{ %Times New Roman sөArial Wingdings Arial NarrowSymbolз Post Modern%Fundamental Graph Theory (Lecture 1)Lecture 1 : What is a graph? Continued KPreliminaries An exampleNeighborhood and Degree The 1st theorem of Graph TheoryOdd and Even graphs SubgraphsBeautiful GraphsMore Beautiful GraphsGeneralized Johnson GraphsPowerPoint ² SupergraphsDirected GraphsComplete Directed GraphsIn-degree and Out-degreeAn Important ExampleWeighted GraphsSigned GraphsLabeled Graphs IsomorphismGeneral SubgraphsGraph Representations Continued KWalks in GraphsThe Number of Walks Distance in GraphsConcluding Remark of Lecture 1  ϥΦr ²]pd vD_Ehlfuhlfu