Math 365 Wednesday 4/10/ & 10.2 Graphs

Transcription

1 Math 365 Wednesda 4/10/ & 10.2 Graphs Eercise 44. (Relations and digraphs) For each the relations in Eercise 43(a), dra the corresponding directed graph here V = {0, 1, 2, 3} and a! b if a b. What properties of the directed graphs correspond to the smmetric, refleie, and transitie properties of the corresponding relations? For the digraphs corresponding to eqialence relations, hat do the eqialence classes look like? Eercise 45. (Applied graphs) Pick for of the Eamples 2 12 in Section 10.1, and qickl smmarize them. What is V? What is E? And hat kind of graph reslts? For eample, in Eample 1, The reslting graph is simple. V = { people } E = {a b a 6= b, and a and b are acqainted } Eercise 46. Let G = a b H = c d (a) In G, hat is the neighborhood of a? What is the neighborhood of b? (b) Calclate the degrees of each erte in G and H. (c) Verif the handshake theorem on G and H.

2 Eercise 47. (a) Dra C 6, W 6 K 6, and K 5,3. (b) Which of the folloing are bipartite? Jstif or anser. (c) Hpercbes are bipartite. (i) The folloing is the 4-cbe: Shade in the ertices that hae an een nmber of 0 s. Eplain h the 4-cbe is bipartite. (ii) Eplain h Q n is bipartite in general. [Hint: Sho that a erte ith an een (respectiel, odd) nmber of 0 s ill neer be adjacent to another erte ith an een (respectiel, odd) nmber of 0 s.]

3 Eercise 48. (a) For each of the folloing pairs of graphs, first list their degree seqences. Then decide hether the are isomorphic or not. If not, sa h. If the are, gie a bijection on the ertices that preseres the edges, and dra the nlabeled graph that represents the corresponding isomorphism class of graphs. (i) (ii) (iii) (i) () (i) (b) Ho man isomorphism classes are there of simple graphs ith 4 ertices? Dra them. (c) Ho man edges does a graph hae if its degree seqence is 4, 3, 3, 2, 2? Dra a graph ith this degree seqence. Can o dra a simple graph ith this seqence? (d) For hich ales of n, m are these graphs reglar? What is the degree? (i) K n (ii) C n (iii) W n (i) Q n () K m,n (e) Ho man ertices does a reglar graph of degree for ith 10 edges hae? (f) Sho that isomorphism of simple graphs is an eqialence relation.

4 A graph is a set of objects, or ertices, together ith a (mlti)set of edges that connect pairs of ertices. (Think driing rotes beteen cities, or social connections beteen people.) Eample: e 2 e 3 e 4 e 5 e 1 Here, the ertices are V t,,,,, and the edges are E te 1, e 2, e 3, e 4, e 5. An edge that connects a erte to itself (like e 5 )iscalledaloop. We sa a erte a is adjacent to a erte b if there is an edge connecting a and b. (Notice that for a generic graph, adjacenc is a smmetric relation, bt is not refleie nor is it transitie.) Classes of graphs: A graph is simple if there are no loops and eer pair of ertices has at most one edge beteen them. Simple! NOT simple! NOT simple!

5 Classes of graphs: A graph is simple if there are no loops and eer pair of ertices has at most one edge beteen them. A graph is a mltigraph if there are no loops, bt there cold be mltiple edges beteen to ertices. Mltigraph! Mltigraph! NOT Mltigraph! Classes of graphs: A graph is simple if there are no loops and eer pair of ertices has at most one edge beteen them. A graph is a mltigraph if there are no loops, bt there cold be mltiple edges beteen to ertices. A graph is a psedograph if there cold be loops or mltiple edges. (This is jst hat e call a graph.) So t psedographs/graphs â t mltigraphs â t simple graphs. (Note: The â smbol is sed here becase, for eample, eer simple graph is a mltigraph, bt there are mltigraphs that are not simple.)

6 Directed graphs A directed graph (also called a digraph or a qier) is a graph, together ith a choice of direction for each edge. (Think flights from one cit to the other, or a flo chart.) For eample, Directed graphs A directed graph (also called a digraph or a qier) is a graph, together ith a choice of direction for each edge. (Think flights from one cit to the other, or a flo chart.) A directed graph is simple if there are no loops and eer pair of ertices has at most one edge in each direction beteen them. Simple! NOT Simple! NOT Simple!

7 Directed graphs A directed graph (also called a digraph or a qier) is a graph, together ith a choice of direction for each edge. (Think flights from one cit to the other, or a flo chart.) A directed graph is simple if there are no loops and eer pair of ertices has at most one edge in each direction beteen them. A directed graph is a directed mltigraph if there cold be loops or mltiple edges. (This is jst hat e call a directed graph) So t directed (mlti)graphs â t directed simple graphs. The book also talks abot mied graphs, here some of the edges are directed and some aren t. We sall take care of this b modeling the non-directed edges ith to directed edges, one in each direction. G e 2 e 3 e 4 e 5 e 1 Npq t, Npq t Npq t Npq t Npq t, Npt, q t,, Npt, q t,,, We sa a erte a is adjacent to a erte b if there is an edge connecting a and b. For eample, in G, is adjacent to and ; is adjacent to ; is adjacent to and. We sa that an edge is incident to a erte if the edge connects to the erte. For eample, in G, e 1 is incident to and ; e 5 is incident to. If to ertices and are adjacent, e e sa that the are neighbors, andthat is in the neighborhood Npq of (and ice-ersa). If A Ñ V,then NpAq PA Npq.

8 G e 2 e 3 e 4 e 5 e 1 Npq t, Npq t Npq t Npq t Npq t, degpq 3 degpq 1 degpq 2 degpq 1 degpq 3 The degree degpq of a erte is the nmber of edge ends attached to. Fact: degpq Npq ; and a graph is simple if and onl if degpq Npq for all P V. H e 2 e 3 e 1 Npq t, Npq t Npq t Npq t Npq t degpq 2 degpq 1 degpq 1 degpq 1 degpq 1 G e 2 e 3 e 4 e 5 e 1 Npq t, Npq t Npq t Npq t Npq t, degpq 3 degpq 1 degpq 2 degpq 1 degpq 3 The degree degpq of a erte is the nmber of edge ends attached to. We call a graph reglar if all the ertices hae the same degree. Theorem (The handshake theorem) In a graph G pv,eq, 2 E ÿ PV deg. Corollar In an graph, there are an een nmber of odd ertices.

9 Graph isomorphisms We sa to graphs G and G 1 are isomorphic if there is a relabeling of the ertices of G that transforms it into G 1. In other ords, there is a bijection f : V Ñ V 1 sch that the indced map on E is a bijection f : E Ñ E 1. For eample, b G a c and G 1 d are isomorphic ia the map (doesn t depend on the draing) a fiñ, b fiñ, c fiñ, d fiñ. Recall: an eqialence relation on a set A is a pairing that is refleie (a a), smmetric (a b i b a), and transitie (a b and b c implies a c). Gien an eqialence relation, an eqialence class is a maimal set of things that are pairise eqialent. Here, if G is the set of all graphs, then G H heneer G is isomorphic to H is an eqialence relation. For an eqialence class of graphs, e dra the associated nlabeled graph. For eample, the eqialence class of graphs corresponding to b b G a c is a c. d d

10 Special graphs Ccles. A ccle C n is the eqialence class of simple graphs on n ertices t 1, 2,..., n so that i is adjacent to i 1 ( 1 is adjacent to n ). eqialence class C 5 1 one graph in the class C Wheels. A heel W n is the ccle C n together ith an additional erte that is adjacent to eer other erte. eqialence class W 5 one graph in the class W Special graphs Complete graphs. The complete graph on n ertices, denoted K n, is the eqialence class of simple graphs on n ertices so that Npq V t for all all P V. For eample, K 1 K 2 K 3 K 4 K 5

11 Bipartite graphs. A graph is bipartite if V can be partitioned into to nonempt sbsets V 1 and V 2 so that no erte in V i is adjacent to an other erte in V i for i 1 or 2. In particlar, for an m n 1, thecomplete bipartite graph K n,m is the class of simple graphs corresponding to the graph ith ertices V V 1 Y V 2,here V 1 t 1,..., n V 2 t 1, m for all i. For eample, Np i q V 2 and Np i q V 1 K 7,3 One a to sho that a graph is bipartite is to color the ertices to di erent colors, so that no to ertices of the same color are adjacent. Bipartite graphs. A graph is bipartite if V can be partitioned into to nonempt sbsets V 1 and V 2 so that no erte in V i is adjacent to an other erte in V i for i 1 or 2. One a to sho that a graph is bipartite is to color the ertices to di erent colors, so that no to ertices of the same color are adjacent. Hpercbes. Let Q n be the graph ith erte set V tbit strings (1 s and 0 s) of length n and edge set E t and di er in eactl one bit Q Q Q Color ertices ith an een nmber of 0 s red.

12 Graph inariants To proe that to graphs are isomorphic, o need to find an isomorphism. To sho that the re not isomorphic, o hae to sho that no isomorphism eists, hich can be harder!so e look for properties of the graphs that are presered b isomorphisms. These are called (graph) inariants. Eample: The nmber of ertices in a graph is an inariant. (If G is isomorphic to H, then there is a bijection beteen their erte sets, so those erte sets mst hae the same size. Conersel, if G and H hae a di erent nmber of ertices, then no sch bijection eits.) For eample, C 5 and C 6 are di erent isomorphism classes. Similarl, the nmber of edges in a graph is an inariant. For eample, C 5 and K 5 are di erent isomorphism classes. Graph inariants To proe that to graphs are isomorphic, o need to find an isomorphism. To sho that the re not isomorphic, o hae to sho that no isomorphism eists, hich can be harder! So e look for properties of the graphs that are presered b isomorphisms. These are called (graph) inariants. Eample: The degree seqence of a graph is the list of degrees of ertices in the graph, gien in decreasing order. For eample, the degree seqence of a b c G e d f is 6, 5, 4, 3, 2, 0. (Again, if the degree seqences of G and H di er, then G fl H. Bt if the degree seqences match, the might be isomorphic, bt the might not be.)

13 Graph inariants For eample, consider the graphs G H 7 8 Both of these graphs hae the degree seqence 3, 3, 2, 2, 1, 1, 1, 1. Bt in G, there s a erte of degree 1 adjacent to a erte of degree 2, here as no erte of degree 1 is adjacent to a erte of degree 2 in H. SoG fl H.