## Question

###### Random graphsGiveI positive integer Llc probability 0 < p < 1 (which could function o n), Lfle T:UId(In graph G(n,p) is graph G = (V,E) o |VI = verGices which every possible edge is present with probability p adl these â‚¬VCItS are indcpendent . The edgc probability could bc COIISGAlIt independent of n (sy could also depend On n like p(n) logln) 9) , but (The sunple space is thc sel of all graphs OL n verlices and Lhe probability of aHLY graph depends On its number of edges_ In Lhis projec

Random graphs GiveI positive integer Llc probability 0 < p < 1 (which could function o n), Lfle T:UId(In graph G(n,p) is graph G = (V,E) o |VI = verGices which every possible edge is present with probability p adl these â‚¬VCItS are indcpendent . The edgc probability could bc COIISGAlIt independent of n (sy could also depend On n like p(n) logln) 9) , but (The sunple space is thc sel of all graphs OL n verlices and Lhe probability of aHLY graph depends On its number of edges_ In Lhis projecL You investigate VATIOIS properties (f random graphs_ Yor results should include: (a) Show that for every < 1/1000 aud p = cn; with probability that Lends to tends Lo infinity; all the cOnnected â‚¬OIponents o G(n, pln)) arc Of size O(log n). (this is actually true for all c < 1, CAIL YOIL prove it?) (b) Show that for eVery 2 1000 and p = c/n; with probability Ghal tends to aS TL tcnds to infinity; Lhere exisbs connecbed COponenl of a linear SIZC. (this is true for all c > 1, CL VOu proC something likc that?) For which p (aS fiicLion p(n) of n) , does the raudom graph G(n, P(n) ) have no isolated vertex (i.e; vertex with no edge incident to it)? Thc type of result yOn should prove is that if p(n) > (1|c)log" Lhen Lhe probability Lhat ((n; p(n)) has a1 isolated vertex tcnds to 0 aS #l Lends G0 infinity (say for fixed while if p(n) log " thcn the probability that G(n, P(n) ) has isolated verlex Lends to 13 It is useful Lo look at the randm valriahle representing Ghc numnbcr of isolated verlices and Lo compute its expectation; variance, ec LolI Show GhaL for p > (1 | c) LheI Ghc probability Lhat G(n, P) is connected tends to #S m Gcncs Lo infinity Writc (ad prove) result stating thc xize: of thc largest clique in the randoI graph G(n; 4). Again Lhis nceds to be formalized. For which valucs of p do we start scing triangles with probability Lhat tends to 1? Extensions YOI (Ould investigate would be when (i.c for which vallues of p) raIldom graph hals at least OHC triangle clique of size 3), OI Wfcn raLlIdom graph is cOected; or Lhe IaLXim (OI IniniIun) degree of raLIIdo graph: There arc countless other questions thal could be considered_ Peoplc: make creeTs Out of this.