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§9.1: Graphs and Graph Models
• Correspond to symmetricCorrespond to symmetric
binary relationsbinary relations RR..
• AA simple graphsimple graph GG=(=(VV,,EE))
consists of:consists of:
– a seta set VV ofof verticesvertices oror nodesnodes ((VV corresponds tocorresponds to
the universe of the relationthe universe of the relation RR),),
– a seta set EE ofof edgesedges // arcsarcs // linkslinks:: unordered pairsunordered pairs ofof
[distinct?][distinct?] elementselements uu,,vv ∈∈ VV, such that, such that uRvuRv..
Visual Representation
of a Simple Graph
§ 9.1 – Graphs and Graph Models](https://image.slidesharecdn.com/graphtheory-150703201136-lva1-app6891/85/Graph-theory-2-320.jpg)
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Adjacency Matrices
• MatrixMatrix AA=[=[aaijij], where], where aaijij isis 11 if {if {vvii,, vvjj} is an} is an
edge ofedge of GG,, 00 otherwise.otherwise.
a b
c d
0001
0011
0101
1110
a b
d c
0212
2110
1103
2030
§ 9.3 – Graph Representations & Isomorphism](https://image.slidesharecdn.com/graphtheory-150703201136-lva1-app6891/85/Graph-theory-29-320.jpg)
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Incidence Matrices
• MatrixMatrix AA=[=[aaijij]]vxe, where, where aaijij isis 11 ifif eejj incidentincident
withwith vvii,, 00 otherwise.otherwise.
b c
d
e
011010
000101
110000
101100
000011a
1
23
4
5
6
§ 9.3 – Graph Representations & Isomorphism](https://image.slidesharecdn.com/graphtheory-150703201136-lva1-app6891/85/Graph-theory-30-320.jpg)
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Counting Paths w Adjacency Matrices
• LetLet AA be the adjacency matrix of graphbe the adjacency matrix of graph GG..
• TheThe numbernumber of paths of lengthof paths of length rr fromfrom vvii toto vvjj
is equal tois equal to ((AArr
))i,ji,j. (The notation (. (The notation (MM))i,ji,j denotesdenotes
mmi,ji,j where [where [mmi,ji,j] =] = MM..))
§ 9.4 – Connectivity](https://image.slidesharecdn.com/graphtheory-150703201136-lva1-app6891/85/Graph-theory-41-320.jpg)
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Graph theory
1. The document defines various types of graphs including simple graphs, multigraphs, pseudographs, directed graphs, and bipartite graphs. It discusses properties like vertices, edges, degrees, adjacency, and special graph structures like complete graphs, cycles, wheels, and n-cubes. 2. Key terms are defined, such as the degree of a vertex, adjacent vertices, and handshaking theorems for both undirected and directed graphs. Special graph structures including complete graphs, cycles, wheels, and n-cubes are also described. 3. Bipartite graphs are introduced as simple graphs whose vertices can be partitioned into two disjoint sets such that every edge connects a vertex in one set to a
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Graph theory
- 1.
- 2. 2 §9.1: Graphs andGraph Models • Correspond to symmetricCorrespond to symmetric binary relationsbinary relations RR.. • AA simple graphsimple graph GG=(=(VV,,EE)) consists of:consists of: – a seta set VV ofof verticesvertices oror nodesnodes ((VV corresponds tocorresponds to the universe of the relationthe universe of the relation RR),), – a seta set EE ofof edgesedges // arcsarcs // linkslinks:: unordered pairsunordered pairs ofof [distinct?][distinct?] elementselements uu,,vv ∈∈ VV, such that, such that uRvuRv.. Visual Representation of a Simple Graph § 9.1 – Graphs and Graph Models
- 3. 3 • LetLet VVbe the set of states in the far-be the set of states in the far- southeastern U.S.:southeastern U.S.: –VV={FL, GA, AL, MS, LA, SC, TN, NC}={FL, GA, AL, MS, LA, SC, TN, NC} • LetLet EE={{={{uu,,vv}|}|uu adjoinsadjoins vv}} ={{FL,GA},{FL,AL},{FL,MS},={{FL,GA},{FL,AL},{FL,MS}, {FL,LA},{GA,AL},{AL,MS},{FL,LA},{GA,AL},{AL,MS}, {MS,LA},{GA,SC},{GA,TN},{MS,LA},{GA,SC},{GA,TN}, {SC,NC},{NC,TN},{MS,TN},{SC,NC},{NC,TN},{MS,TN}, {MS,AL}}{MS,AL}} Example of a Simple Graph TN ALMS LA SC GA FL NC § 9.1 – Graphs and Graph Models
- 4. 4 Multigraphs • Like simplegraphs, butLike simple graphs, but there may bethere may be moremore than onethan one edge connecting two given nodesedge connecting two given nodes.. • AA multigraphmultigraph GG=(=(VV,, EE,, ff ) consists of a set) consists of a set VV of vertices, a setof vertices, a set EE of edges (as primitiveof edges (as primitive objects), and a functionobjects), and a function ff::EE→→{{{{uu,,vv}|}|uu,,vv∈∈VV ∧∧ uu≠≠vv}.}. • E.g., nodes are cities, edgesE.g., nodes are cities, edges are segments of major highways.are segments of major highways. Parallel edges § 9.1 – Graphs and Graph Models
- 5. 5 Pseudographs • Like amultigraph, butLike a multigraph, but edges connecting aedges connecting a node to itself are allowednode to itself are allowed.. • AA pseudographpseudograph GG=(=(VV,, EE,, ff ) where) where ff::EE→→{{{{uu,,vv}|}|uu,,vv∈∈VV}. Edge}. Edge ee∈∈EE is ais a looploop ifif ff((ee)={)={uu,,uu}={}={uu}}.. • E.g.E.g., nodes are campsites, nodes are campsites in a state park, edges arein a state park, edges are hiking trails through the woods.hiking trails through the woods. § 9.1 – Graphs and Graph Models loops
- 6. 6 Directed Graphs • Correspondto arbitrary binary relationsCorrespond to arbitrary binary relations RR,, which need not be symmetric.which need not be symmetric. • AA directed graphdirected graph ((VV,,EE) consists of a set of) consists of a set of verticesvertices VV and a binary relationand a binary relation EE onon VV.. • E.g.E.g.:: VV = people,= people, EE={(={(xx,,yy) |) | xx lovesloves yy}} • ee=(=(uu,,vv) ,) ,uu,,vv∈∈VV § 9.1 – Graphs and Graph Models
- 7. 7 Directed Multigraphs • Likedirected graphs, but there may be moreLike directed graphs, but there may be more than one arc from a node to another.than one arc from a node to another. • AA directed multigraphdirected multigraph GG=(=(VV,, EE,, ff ) consists) consists of a setof a set VV of vertices, a setof vertices, a set EE of edges, and aof edges, and a functionfunction ff::EE→→VV××VV.. • E.g.,E.g., VV=web pages,=web pages, EE=hyperlinks.=hyperlinks. The WWW isThe WWW is a directed multigraph...a directed multigraph... § 9.1 – Graphs and Graph Models
- 8. 8 Types of Graphs:Summary • Summary of the book’s definitions.Summary of the book’s definitions. • Keep in mind this terminology is not fullyKeep in mind this terminology is not fully standardized...standardized... Term Edge type Multiple edges ok? Self- loops ok? Simple graph Undir. No No Multigraph Undir. Yes No Pseudograph Undir. Yes Yes Directed graph Directed No Yes Directed multigraph Directed Yes Yes § 9.1 – Graphs and Graph Models
- 9. 9 §9.2: Graph Terminology •Adjacent, connects, endpoints, degree,Adjacent, connects, endpoints, degree, initial, terminal, in-degree, out-degree,initial, terminal, in-degree, out-degree, complete, cycles, wheels, n-cubes, bipartite,complete, cycles, wheels, n-cubes, bipartite, subgraph, union.subgraph, union. § 9.2 – Graph Terminology
- 10. 10 Adjacency LetLet GG bean undirected graph with edge setbe an undirected graph with edge set EE.. LetLet ee∈∈EE be (or map to) the pair {be (or map to) the pair {uu,,vv}. Then}. Then we say:we say: • uu,, vv areare adjacentadjacent // neighborsneighbors // connectedconnected.. • EdgeEdge ee isis incident withincident with verticesvertices uu andand vv.. • EdgeEdge ee connectsconnects uu andand vv.. • VerticesVertices uu andand vv areare endpointsendpoints of edgeof edge ee.. § 9.2 – Graph Terminology
- 11. 11 Degree of aVertex • LetLet GG be an undirected graph,be an undirected graph, vv∈∈VV a vertex.a vertex. • TheThe degreedegree ofof vv, deg(, deg(vv), is its number of), is its number of incident edges. (Except that any self-loopsincident edges. (Except that any self-loops are counted twice.)are counted twice.) • A vertex with degree 0 isA vertex with degree 0 is isolatedisolated.. • A vertex of degree 1 isA vertex of degree 1 is pendantpendant.. § 9.2 – Graph Terminology
- 12. 12 Handshaking Theorem • LetLetGG be an undirected (simple, multi-, orbe an undirected (simple, multi-, or pseudo-) graph with vertex setpseudo-) graph with vertex set VV and edgeand edge setset EE. Then. Then • Corollary: Any undirected graph has anCorollary: Any undirected graph has an even number of vertices of odd degreeeven number of vertices of odd degree.. Ev Vv 2)deg( =∑∈ § 9.2 – Graph Terminology
- 13. 13 Directed Adjacency • LetLetGG be a directed (possibly multi-) graph,be a directed (possibly multi-) graph, and letand let ee be an edge ofbe an edge of GG that is (or maps to)that is (or maps to) ((uu,,vv). Then we say:). Then we say: – uu isis adjacent toadjacent to vv,, vv isis adjacent fromadjacent from uu – ee comes fromcomes from uu,, ee goes togoes to vv.. – e connects u to ve connects u to v,, e goes from u to ve goes from u to v – thethe initial vertexinitial vertex ofof ee isis uu – thethe terminal vertexterminal vertex ofof ee isis vv § 9.2 – Graph Terminology
- 14. 14 Directed Degree • LetLetGG be a directed graph,be a directed graph, vv a vertex ofa vertex of GG.. – TheThe in-degreein-degree ofof vv, deg, deg−− ((vv), is the number of), is the number of edges going toedges going to vv.. – TheThe out-degreeout-degree ofof vv, deg, deg++ ((vv), is the number of), is the number of edges coming fromedges coming from vv.. – TheThe degreedegree ofof vv, deg(, deg(vv))≡≡degdeg−− ((vv)+deg)+deg++ ((vv), is the), is the sum ofsum of vv’s in-degree and out-degree.’s in-degree and out-degree. § 9.2 – Graph Terminology
- 15. 15 Directed Handshaking Theorem •LetLet GG be a directed (possibly multi-) graphbe a directed (possibly multi-) graph with vertex setwith vertex set VV and edge setand edge set EE. Then:. Then: • Note that the degree of a node is unchangedNote that the degree of a node is unchanged by whether we consider its edges to beby whether we consider its edges to be directed or undirected.directed or undirected. Evvv VvVvVv === ∑∑∑ ∈∈ + ∈ − )deg( 2 1 )(deg)(deg § 9.2 – Graph Terminology
- 16. 16 Special Graph Structures Specialcases of undirected graph structures:Special cases of undirected graph structures: • Complete graphsComplete graphs KKnn • CyclesCycles CCnn • WheelsWheels WWnn • nn-Cubes-Cubes QQnn • Bipartite graphsBipartite graphs • Complete bipartite graphsComplete bipartite graphs KKmm,,nn § 9.2 – Graph Terminology
- 17. 17 Complete Graphs • ForanyFor any nn∈∈NN, a, a complete graphcomplete graph onon nn vertices,vertices, KKnn, is a simple graph with, is a simple graph with nn nodesnodes in which every node is adjacent to everyin which every node is adjacent to every other node:other node: ∀∀uu,,vv∈∈VV:: uu≠≠vv↔↔{{uu,,vv}}∈∈EE.. Note that Kn has edges.2 )1(1 1 − =∑ − = nn i n i K1 K2 K3 K5 K6 K4 § 9.2 – Graph Terminology
- 18. 18 Cycles • For anyForany nn≥≥3, a3, a cyclecycle onon nn vertices,vertices, CCnn, is a, is a simple graph wheresimple graph where VV={={vv11,,vv22,… ,,… ,vvnn} and} and EE={{={{vv11,,vv22},{},{vv22,,vv33},…,{},…,{vvnn−−11,,vvnn},{},{vvnn,,vv11}}.}}. C3 C4 C5 C6 C7 C8 How many edges are there in Cn? § 9.2 – Graph Terminology
- 19. 19 Wheels • For anyForany nn≥≥3, a3, a wheelwheel WWnn, is a simple graph, is a simple graph obtained by taking the cycleobtained by taking the cycle CCnn and addingand adding one extra vertexone extra vertex vvhubhub andand nn extra edgesextra edges {{{{vvhubhub,,vv11}, {}, {vvhubhub,,vv22},…,{},…,{vvhubhub,,vvnn}}.}}. How many edges are there in Wn? W3 W5 W6 W7 W8 W4 § 9.2 – Graph Terminology
- 20. 20 n-cubes (hypercubes) • ForanyFor any nn∈∈NN, the hypercube, the hypercube QQnn is a simpleis a simple graph consisting of two copies ofgraph consisting of two copies of QQnn-1-1 connected together at corresponding nodes.connected together at corresponding nodes. QQ00 has 1 node.has 1 node. Q0 Q1 Q2 Q3 Q4 Number of vertices: 2n . Number of edges:Exercise to try! § 9.2 – Graph Terminology
- 21. 21 n-cubes (hypercubes) • ForanyFor any nn∈∈NN, the hypercube, the hypercube QQnn can becan be defined recursively as follows:defined recursively as follows: – QQ00={{={{vv00},},∅∅} (one node and no edges)} (one node and no edges) – For anyFor any nn∈∈NN, if Q, if Qnn==((VV,,EE), where), where VV={={vv11,…,,…,vvaa}} andand EE={={ee11,…,,…,eebb}, then}, then QQnn+1+1=(=(VV∪∪{{vv11´,…,´,…,vvaa´},´}, EE∪∪{{ee11´,…,´,…,eebb´}´}∪∪{{{{vv11,,vv11´},{´},{vv22,,vv22´},…,´},…, {{vvaa,,vvaa´}}) where´}}) where vv11´,…,´,…,vvaa´ are new vertices, and´ are new vertices, and where ifwhere if eeii={={vvjj,,vvkk} then} then eeii´={´={vvjj´,´,vvkk´}.´}. § 9.2 – Graph Terminology
- 22. 22 Bipartite Graphs • AAsimple graphsimple graph GG is calledis called bipartitebipartite if itsif its vertex setvertex set VV can be partitioned into twocan be partitioned into two disjoint setsdisjoint sets VV1 andand VV2 such thatsuch that every edgeevery edge in the graph connects a vertex inin the graph connects a vertex in VV1 and aand a vertex invertex in VV2(so that no edge in(so that no edge in GG connectsconnects either two vertices ineither two vertices in VV1 or two vertices inor two vertices in VV2)) § 9.2 – Graph Terminology
- 23. 23 Complete Bipartite Graphs •TheThe complete bipartite graphcomplete bipartite graph KKm,n that has itsthat has its vertex set partitioned into two subsets ofvertex set partitioned into two subsets of mm andand nn vertices, respectively. There is anvertices, respectively. There is an edge between two verticesedge between two vertices iffiff one vertex isone vertex is in the first subset and the other is in thein the first subset and the other is in the second subsetsecond subset.. § 9.2 – Graph Terminology
- 24. 24 Subgraphs • A subgraphof a graphA subgraph of a graph GG=(=(VV,,EE) is a graph) is a graph HH=(=(WW,,FF) where) where WW⊆⊆VV andand FF⊆⊆EE.. G H § 9.2 – Graph Terminology
- 25. 25 Graph Unions • TheTheunionunion GG11∪∪GG22 of two simple graphsof two simple graphs GG11=(=(VV11,, EE11) and) and GG22=(=(VV22,,EE22) is the simple) is the simple graph (graph (VV11∪∪VV22,, EE11∪∪EE22).). § 9.2 – Graph Terminology
- 26. 26 §9.3: Graph Representations& Isomorphism • Graph representations:Graph representations: – Adjacency lists.Adjacency lists. – Adjacency matrices.Adjacency matrices. – Incidence matrices.Incidence matrices. • Graph isomorphism:Graph isomorphism: – Two graphs are isomorphic iff they are identicalTwo graphs are isomorphic iff they are identical except for theirexcept for their node namesnode names.. § 9.3 – Graph Representations & Isomorphism
- 27. 27 Adjacency Lists • Atable with 1 row per vertex, listing itsA table with 1 row per vertex, listing its adjacent vertices.adjacent vertices. a b dc f e Vertex Adjacent Vertices a b c d e f § 9.3 – Graph Representations & Isomorphism
- 28. 28 Directed Adjacency Lists •1 row per node, listing the terminal nodes of1 row per node, listing the terminal nodes of each edge incident from that node.each edge incident from that node. § 9.3 – Graph Representations & Isomorphism a b dc f e Initial Vertex Terminal Vertices a b c d e f
- 29. 29 Adjacency Matrices • MatrixMatrixAA=[=[aaijij], where], where aaijij isis 11 if {if {vvii,, vvjj} is an} is an edge ofedge of GG,, 00 otherwise.otherwise. a b c d 0001 0011 0101 1110 a b d c 0212 2110 1103 2030 § 9.3 – Graph Representations & Isomorphism
- 30. 30 Incidence Matrices • MatrixMatrixAA=[=[aaijij]]vxe, where, where aaijij isis 11 ifif eejj incidentincident withwith vvii,, 00 otherwise.otherwise. b c d e 011010 000101 110000 101100 000011a 1 23 4 5 6 § 9.3 – Graph Representations & Isomorphism
- 31. 31 Graph Isomorphism • Formaldefinition:Formal definition: – Simple graphsSimple graphs GG11=(=(VV11,, EE11) and) and GG22=(=(VV22,, EE22) are) are isomorphicisomorphic iffiff ∃∃ a bijectiona bijection ff::VV11→→VV22 such thatsuch that ∀∀ aa,,bb∈∈VV11,, aa andand bb are adjacent inare adjacent in GG11 iffiff ff((aa) and) and ff((bb) are adjacent in) are adjacent in GG22.. – ff is the “renaming” function that makes the twois the “renaming” function that makes the two graphs identical.graphs identical. – Definition can easily be extended to other typesDefinition can easily be extended to other types of graphs.of graphs. § 9.3 – Graph Representations & Isomorphism
- 32. 32 Graph Invariants underIsomorphism NecessaryNecessary but notbut not sufficientsufficient conditions forconditions for GG11=(=(VV11,, EE11) to be isomorphic to) to be isomorphic to GG22=(=(VV22,, EE22):): – ||VV1|=|1|=|VV2|, |2|, |EE1|=|1|=|EE2|.2|. – The number of vertices with degreeThe number of vertices with degree nn is theis the same in both graphs.same in both graphs. – For every proper subgraphFor every proper subgraph gg of one graph, thereof one graph, there is a proper subgraph of the other graph that isis a proper subgraph of the other graph that is isomorphic toisomorphic to gg.. § 9.3 – Graph Representations & Isomorphism
- 33. 33 Isomorphism Example • Ifisomorphic, label the 2nd graph to showIf isomorphic, label the 2nd graph to show the isomorphism, else identify difference.the isomorphism, else identify difference. a b cd e f b d a e fc § 9.3 – Graph Representations & Isomorphism
- 34. 34 Are These Isomorphic? •If isomorphic, label the 2nd graph to showIf isomorphic, label the 2nd graph to show the isomorphism, else identify difference.the isomorphism, else identify difference. a b c d e * Same # of vertices * Same # of edges * Different # of verts of degree 2! (1 vs 3) § 9.3 – Graph Representations & Isomorphism
- 35. 35 §9.4: Connectivity • Inan undirected graph, aIn an undirected graph, a path of length npath of length n from u to vfrom u to v is a sequence of adjacent edgesis a sequence of adjacent edges going from vertexgoing from vertex uu to vertexto vertex vv.. • A path is aA path is a circuitcircuit ifif u=vu=v.. • A pathA path traversestraverses the vertices along it.the vertices along it. • A path isA path is simplesimple ifif it contains no edge moreit contains no edge more than oncethan once.. § 9.4 – Connectivity
- 36. 36 Paths in DirectedGraphs • Same as in undirected graphs, butSame as in undirected graphs, but the paththe path must go in the directionmust go in the direction of the arrows.of the arrows. § 9.4 – Connectivity
- 37. 37 Connectedness • An undirectedgraph isAn undirected graph is connectedconnected iffiff there is athere is a path between every pair of distinct vertices in thepath between every pair of distinct vertices in the graphgraph.. • Theorem: There is aTheorem: There is a simplesimple path between any pairpath between any pair of distinct vertices in a connected undirectedof distinct vertices in a connected undirected graph.graph. • Connected componentConnected component: connected subgraph: connected subgraph • AA cut vertexcut vertex oror cut edgecut edge separates 1 connectedseparates 1 connected component into 2 if removed.component into 2 if removed. § 9.4 – Connectivity
- 38. 38 Directed Connectedness • Adirected graph isA directed graph is strongly connectedstrongly connected iffiff there is a directed paththere is a directed path fromfrom aa toto bb and fromand from bb toto aa for any two verticesfor any two vertices aa andand bb.. • It isIt is weakly connectedweakly connected iff the underlyingiff the underlying undirectedundirected graph (graph (i.e.i.e., with edge directions, with edge directions removed) is connected.removed) is connected. • NoteNote stronglystrongly impliesimplies weaklyweakly but not vice-but not vice- versa.versa. § 9.4 – Connectivity
- 39. 39 Example • Are thegraphsAre the graphs GG andand HH stronglystrongly connectedconnected?? Are theyAre they weakly connectedweakly connected?? a b c de a b c de G H § 9.4 – Connectivity
- 40. 40 Paths & Isomorphism •Note that connectedness, and the existenceNote that connectedness, and the existence of a circuit orof a circuit or simple circuit of lengthsimple circuit of length kk areare graph invariantsgraph invariants with respect towith respect to isomorphismisomorphism.. a b c de f m n o pq r IsomorphicIsomorphic?? § 9.4 – Connectivity
- 41. 41 Counting Paths wAdjacency Matrices • LetLet AA be the adjacency matrix of graphbe the adjacency matrix of graph GG.. • TheThe numbernumber of paths of lengthof paths of length rr fromfrom vvii toto vvjj is equal tois equal to ((AArr ))i,ji,j. (The notation (. (The notation (MM))i,ji,j denotesdenotes mmi,ji,j where [where [mmi,ji,j] =] = MM..)) § 9.4 – Connectivity
- 42. 42 Example • How manypaths of length 4 fromHow many paths of length 4 from aa toto dd?? a b cd = 0110 1001 1001 0110 A =4 A § 9.4 – Connectivity