Graphs Recall from last time: A graph is a set of objects, or vertices, together with a (multi)set of edges that connect pairs of vertices.

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1 Graphs Reall from last time: A graph is a set of ojets, or erties, together ith a (mlti)set of eges that onnet pairs of erties. Eample: e 2 e 3 e 4 e 5 e 1 Here, the erties are V t,,,,, an the eges are E te 1, e 2, e 3, e 4, e 5. We sa a erte a is ajaent to a erte if there is an ege onneting a an. (Notie that for a generi graph, ajaen is a smmetri relation, t is not refleie nor is it transitie.) We sa that an ege is inient to a erte if the ege onnets to the erte. For eample, in G, e 2 is inient to an ; e 5 is inient to. Neighors an egree G e 2 e 3 e 4 e 5 e 1 Npq t, Npq t Npq t Npq t Npq t, egpq 2 egpq 1 egpq 1 egpq 1 egpq 1 If to erties an are ajaent, e e sa that the are neighors, anthat is in the neighorhoo Npq of (an ie-ersa). The egree egpq of a erte is the nmer of ege ens attahe to. We all a graph reglar if all the erties hae the same egree.

2 Simple graphs A graph is simple if there are no loops an eer pair of erties has at most one ege eteen them. Simple! NOT simple! NOT simple!

3 Graph isomorphisms We sa to graphs G an G 1 are isomorphi (ritten G G 1 )if there is a relaeling of the erties of G that transforms it into G 1. In other ors, there is a ijetion f : V Ñ V 1 sh that the ine map on E is a ijetion f : E Ñ E 1. For eample, G a an G 1 are isomorphi ia the map a fiñ, fiñ, fiñ, fiñ. Graph isomorphisms We sa to graphs G an G 1 are isomorphi if there is a relaeling of the erties of G that transforms it into G 1. In other ors, there is a ijetion f : V Ñ V 1 sh that the ine map on E is a ijetion f : E Ñ E 1. For eample, G a an G 1 are isomorphi ia the map (oesn t epen on the raing) a fiñ, fiñ, fiñ, fiñ.

4 Reall: an eqialene relation on a set A is a pairing that is refleie (a a), smmetri (a i a), an transitie (a an implies a ). Gien an eqialene relation, an eqialene lass 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 isomorphi to H is an eqialene relation. For an eqialene lass of graphs, e ra the assoiate nlaele graph. For eample, the eqialene lass of graphs orresponing to G a is a. Speial graphs Cles. A le C n is the eqialene lass of simple graphs on n erties t 1, 2,..., n so that i is ajaent to i 1 ( 1 is ajaent to n ). eqialene lass C 5 1 one graph in the lass C Wheels. A heel W n is the le C n together ith an aitional erte that is ajaent to eer other erte. eqialene lass W 5 one graph in the lass W

5 Speial graphs Complete graphs. The omplete graph on n erties, enote K n, is the eqialene lass of simple graphs on n erties so that Npq V t for all all P V. For eample, K 1 K 2 K 3 K 4 K 5 Bipartite graphs. A graph is ipartite if V an e partitione into to nonempt ssets V 1 an V 2 so that no erte in V i is ajaent to an other erte in V i for i 1 or 2. In partilar, for an m n 1, theomplete ipartite graph K n,m is the lass of simple graphs orresponing to the graph ith erties 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 an Np i q V 1 K 7,3 One a to sho that a graph is ipartite is to olor the erties to i erent olors, so that no to erties of the same olor are ajaent.

6 Bipartite graphs. A graph is ipartite if V an e partitione into to nonempt ssets V 1 an V 2 so that no erte in V i is ajaent to an other erte in V i for i 1 or 2. One a to sho that a graph is ipartite is to olor the erties to i erent olors, so that no to erties of the same olor are ajaent. Hperes. Let Q n e the graph ith erte set V tit strings (1 s an 0 s) of length n an ege set E t an i er in eatl one it Q Q Q Color erties ith an een nmer of 0 s re. Graph inariants To proe that to graphs are isomorphi, o nee to fin an isomorphism. To sho that the re not isomorphi, o hae to sho that no isomorphism eists, hih an e harer!so e look for properties of the graphs that are presere isomorphisms. These are alle (graph) inariants. Eample: The nmer of erties in a graph is an inariant. (If G is isomorphi to H, then there is a ijetion eteen their erte sets, so those erte sets mst hae the same size. Conersel, if G an H hae a i erent nmer of erties, then no sh ijetion eits.) For eample, C 5 an C 6 are i erent isomorphism lasses. Similarl, the nmer of eges in a graph is an inariant. For eample, C 5 an K 5 are i erent isomorphism lasses.

7 Graph inariants To proe that to graphs are isomorphi, o nee to fin an isomorphism. To sho that the re not isomorphi, o hae to sho that no isomorphism eists, hih an e harer! So e look for properties of the graphs that are presere isomorphisms. These are alle (graph) inariants. Eample: The egree seqene of a graph is the list of egrees of erties in the graph, gien in ereasing orer. For eample, the egree seqene of a G e f is 6, 5, 4, 3, 2, 0. (Again, if the egree seqenes of G an H i er, then G fl H. Bt if the egree seqenes math, the might e isomorphi, t the might not e.) Graph inariants For eample, onsier the graphs G H 7 8 Both of these graphs hae the egree seqene 3, 3, 2, 2, 1, 1, 1, 1. Bt in G, there s a erte of egree 1 ajaent to a erte of egree 2, here as no erte of egree 1 is ajaent to a erte of egree 2 in H. SoG fl H.