Edge Count Clique Alg (EC): A graph C is a clique iff |EC||PUC|=COMB(|VC|,2)|VC|!/((|VC|-2)!2!) SubGraph existence thm (SGE): (VC,EC) is a k-clique iff every induced k-1 subgraph, (VD,ED) is a (k-1)-clique. Apriori Clique Mining Alg (AP): finds all cliques in a graph. For Clique-Mining we can use an ARM-Apriori-like downward closure property: CSkkCliqueSet, CCSk+1Candidatek+1CliqueSet. By SGE, CCSk+1= all s of CSk pairs having k-1 common vertices. Let CCCSk+1 be a union of two k-cliques with k-1 common vertices. Let v and w be the kth vertices (different) of the two k-cliques, then CCSk+1 iff (PE)(v,w)=1. Breadth-1st Clique Alg: CLQK=all Kcliques. Find CLQ3 using CS0. Induction theorem: A Kclique and 3clique that share an edge form a (K+1)clique iff all K-2 edges from the non-shared Kclique vertices to the non-shared 3clique vertex exist. Next find CLQ4, then CLQ5, … Depth-1st Clique Alg: Find a Largest Maximal Clique v. Let (x,y)CLQ3pTree(v,w). If (x,y)E and Count(NewPtSet(v,w,x,y)CLQ3pTree(v,w)&CLQ3pTree(x,y)) is: 0, the 4 vertices form a maximal 4Clique (i.e., v,w,x,y). 1, the 5 vertices form a maximal 5Clique (i.e., v,w,x,y and the NewPt) 2, the 6 vertices form a maximal 6Clique if the NewPair is an edge, else they form 2 maximal 5Cliques. 3, the 7 vertices form a maximal 7Clique if each of the 3 NewPairs is an edge, elseif 1 or 2 of the NewPairs are edges then each of the 6VertexSets (vwxy + 2 EdgeEndpts) form Max6Clique, elseif 0 NewPairs is an edge, then each 5VertexSet (vwxy + 1 NewVertex) forms a maximal 5Clique…. Theorem:  hCliqueNewPtSet, those h vertices together with v,w,x,y form a maximal h+4Clique, where NPS(v,w,x,y)=CLQ3(v,w)&CLQ3(x,y). We can determine if each maximal kClique found is a “Largest” from counts (or find them all) but determining “Largest” early can save time (can move on to another v immediately). E.g., if there aren’t enough siblings left or a large enough 1-count among CLQ3pTrees… Bipartite Clique Mining Algorithm finds many of the Maximal Cliques (MCLQs) in a bipartite graph at a low cost (only selected pairwise ANDs). Each LETTERpTree is a MCLQ unless there are pairwise ANDs with the same count. If so, all LETTERs involved in those pairwise ANDs form a MCLQ with the set of NUMBERS making up that common count (The same is true for NUMBERpTree.). More simply put:  pTree, A, the AND of all pairwise ANDs. A&B, B with Ct(A&B)=Ct(A) is a MCLQ (including A as one such B takes care of case when there are no other Bs s.t. Ct(A&B)=Ct(A)). There is potential for a k-plex [k-core] mining algorithm in this vein. Instead of Ct(A&B)=Ct(A), we would consider. E.g., Ct(A&B)=Ct(A)-1. Each such pTree, C, would be missing just 1vertex (Therefore 1 edge). Thus taking any MCLQ found as above, ANDing in CpTree would produce a 1-plex. ANDing in k such C’s would produce a k-plex. In fact, suppose we have produced a k-plex in such a manner, then ANDing in any C with Ct(C)=Ct(A)-h would produce a (K+h)-plex. &i=1..nAi is a [i=1..nCt(Ai)]-Core

Breadth First Bipartite Max Clique Mine on G9 Each LETTERpTree is a MCLQ unless there are pairwise ANDs with the same count. B A 1 2 C 3 D E F G H I J K L N M C B 1 3 D 2 E F G H I A J K L N M E C 1 6 F 5 G 4 H I 2 L C D 1 4 E F 3 G H I 2 J K L F E 1 6 G H 7 I 3 E F 1 6 G 4 H 7 I H G 1 8 I 5 G H 1 8 I 9 H I 1 9 L J 1 5 C E F G 3 H 4 I K C D E F 1 G 2 H I 3 J L L C E F 1 G 4 H 5 I G M 1 2 I 3 J L A-F N K H G N 1 2 I 3 J L A-F M K H A 1 3 B 1 3 C 1 6 D 1 4 E 1 8 F 1 8 G 1 a H 1 e I 1 c J 1 5 K 1 4 L 1 6 M 1 3 N 1 3 1 2 3 4 5 6 7 8 9 a b v d e f g h i 124-ACEFH is a (3,5)MCLQ 345-CDEG is a (3,4)MCLQ 359-EGHI is a (3,4)MCLQ (1,3,8,9,10,11,12,13,16)-HI is a (9,2)MCLQ 123-BCEFH is a (3,5)MCLQ 12346-CEFH is a (5,4)MCLQ (10,13)-GHIL is a (2,4)MCLQ (14,17,18)-IK is a (3,2)MCLQ 123456-CE is a (6,2)MCLQ 2347-EFGH is a (4,4)MCLQ (11,12,13)-HIJL is a (3,4)MCLQ 1345-CDE is a (4,3)MCLQ 78-EFG is a (2,3)MCLQ (12,13)-HIJLMN is a (2,6)MCLQ (11,12,13,14,15)-JL is a (5,2)MCLQ (12,13,14)-MIJLN is a (3,5)MCLQ The #pTrees that are (1,many)MCLQs: 1,2,3,4,13,14,15 The LETpTrees that are (many,1)MCLQs: E,F,G,H,I,K,L Each #pTree is a MCLQ unless  pairwise ANDs with same Ct, then those numbers together with common letters form a MCLQ 1 3 7 e 2 2 1 6 3 4 d e 3 1 7 e 4 1 6 3 e 2 d 5 1 3 2 4 6 7 9 a b c d e f 6 1 4 2 3 5 7 9 a b c d e f 7 1 3 2 4 5 6 9 a b c d e f 8 1 2 3 4 5 6 7 9 a b c d e f 1 8 2 1 7 3 1 8 4 1 7 5 1 4 6 1 4 7 1 4 8 1 3 9 1 4 a 1 4 b 1 4 c 1 6 d 1 7 e 1 8 f 1 5 g 1 2 h 1 2 i 1 2 B A C D E F G H I J K L M N 9 1 3 2 4 5 6 7 a b c d e f b a 1 3 c d 4 e f g 2 h i 5 6 7 9 e b 1 3 f g 2 h i 4 5 6 7 9 a c d c 1 2 3 4 d 6 e 5 d 1 2 3 4 e 6 e 1 2 3 4 f 1 2 3 4 c d e g 2 1 3 4 5 6 7 8 9 a b c d e f h i 1 2 3 4 5 6 7 8 9 a b c d e f g h i h or i

G9 H I F E C K A L D M J B G N 124-ACEFH is a (3,5)MCLQ 345-CDEG is a (3,4)MCLQ 359-EGHI is a (3,4)MCLQ (1,3,8,9,10,11,12,13,16)-HI is a (9,2)MCLQ 123-BCEFH is a (3,5)MCLQ 12346-CEFH is a (5,4)MCLQ (10,13)-GHIL is a (2,4)MCLQ (14,17,18)-IK is a (3,2)MCLQ 123456-CE is a (6,2)MCLQ 2347-EFGH is a (4,4)MCLQ (11,12,13)-HIJL is a (3,4)MCLQ 1345-CDE is a (4,3)MCLQ (11,12,13,14,15)-JL is a (5,2)MCLQ 78-EFG is a (2,3)MCLQ (12,13)-HIJLMN is a (2,6)MCLQ (12,13,14)-MIJLN is a (3,5)MCLQ G9 A B C D E F G H I J K L M N Bipartite graph of the Southern Women Event Participation. Women are numbers, events are letters. Or Investors are numbers and stocks are letters.

Bipartite Max Clique Mining for the rest of the MCLQs on G9 What we have NOT yet found is MCLQs that do not include a full pTree A C E F H 1 3 B C E F H 1 3 D C E 1 4 B A 1 2 D A 1 2 G A 1 2 A I 1 A J A K A L A M A N A B 1 2 D B 1 2 G B 1 2 I B 1 2 B J B K B L B M B N E C 1 6 F C 1 5 G C 1 4 H C 1 4 I C 1 2 C L F D 1 3 G D 1 3 H D 1 3 I D 1 2 D J D K D L F E 1 6 G E 1 6 H E 1 7 I E 1 3 E F 1 6 G F 1 4 H F 1 7 I F 1 4 H G 1 8 I G 1 5 G H 1 8 I H 1 9 H I 1 9 J C J E J F 1 G J 1 3 H J 1 4 J I 1 L J 1 5 K C K D K E K F 1 G K 1 2 K H 1 I K 1 3 J K 1 2 L K 1 2 L C L E L F 1 G L 1 4 H L 1 5 I L 1 5 M A-F G M 1 2 H M 1 2 I M 1 3 J M 1 3 M K 1 L M 1 3 N M 1 3 N A-F G N 1 2 H N 1 2 I N 1 3 J N 1 3 N K 1 L N 1 3 M N 1 3 A 1 3 B 1 3 C 1 6 D 1 4 E 1 8 F 1 8 G 1 a H 1 e I 1 c J 1 5 K 1 4 L 1 6 M 1 3 N 1 3 A C E F H D G 1 2 B A B C E F H 1 2 D G I B C D E F H 1 2 G I B C E F H 1 2 I C D E F H 1 3 G E C F 1 5 14-ABCEFGH is a (2,7)MCLQ 12-ABCEFH is a (2,6)MCLQ 13-BCDEFHI is a (2,7)MCLQ 12346-CEF is a (5,3)MCLQ 134-CDEFH is a (3,5)MCLQ 345-CDEFG is a (3,5)MCLQ

Depth First Bipartite Max Clique Mine on G9 Find all MCLQ(A) 1 2 C A 1 3 D A 1 2 E A 1 3 F A 1 3 G A 1 2 H A 1 3 A I 1 A J A K A L A M A N E B 1 3 F G 2 H I A J K L N M C D E C 1 6 F 5 G 4 H I 2 L C D 1 4 E F 3 G H I 2 J K L F E 1 6 G H 7 I 3 E F 1 6 G 4 H 7 I H G 1 8 I 5 G H 1 8 I 9 H I 1 9 L J 1 5 C E F G 3 H 4 I K C D E F 1 G 2 H I 3 J L L C E F 1 G 4 H 5 I G M 1 2 I 3 J L A-F N K H G N 1 2 I 3 J L A-F M K H A 1 3 B 1 3 C 1 6 D 1 4 E 1 8 F 1 8 G 1 a H 1 e I 1 c J 1 5 K 1 4 L 1 6 M 1 3 N 1 3 The ApTreeMCLQ(A) unless there are A&XpTrees with Ct(A&X)=Ct(A). Ct(A)=Ct(AC)=Ct(AE)=Ct(AF)=Ct(AH)=3, so ACEFH-124MCLQ(A). Next check for MCLQs with Ct=Ct(A)-1=2. We have 2# CLQs: ACEFH-12, ACEFH-14 and ACEFH-24. Each Ct(A&XpTree)=2=CtA-1 expands one of these 3. (namely AB-12 AD-14 AG-24), Each expanded CLQ is maximal. We get new 2# MCLQ(A): ABCEFH-12 ACDEFH-14 ACEFGH-24 Next check MCLQs with Ct=Ct(A)-2=1. When we reach Ct=1 we simply check the #pTrees containing A (e.g., 1,2,4) for maximal with the 1st step. The BpTreeMCLQ(B) unless Ct(B&X)=Ct(B). 3=Ct(B)=Ct(BC)=Ct(BE)=Ct(BF)=Ct(BH), so BCEFH-123MCLQ(B). Next check MCLQs w Ct=Ct(B)-1=2. We have 2# CLQs: BCEFH-12, BCEFH-13, BCEFH-23 . Each Ct(B&X)=2 expands one of these 3. (namely BA-12 BD-13 BG-23 ), Each expanded CLQ is max. We get 2# MCLQ(B): ABCEFH-12 BCDEFH-13 BCEFGH-23, but ABCEFH-12 not new. Next MCLQs w Ct=Ct(B)-2=1. Check #pTrees B (e.g., 1,2,3) w 1st step (only 3 is new). CMCLQ(C) unless Ct(C&X)=Ct(C). 6=Ct(C)=Ct(CE) so CE-123456MCLQ(C). Next check MCLQs w Ct=Ct(C)-1=5. We have 5# CLQs: CE-12345 CE-12346 CE-12356 CE-12456 CE-13456 CE-23456 Each Ct(C&X)=5 expands one of these, namely CF-12346 (expands to CEF-12346) The Ct(C&X)=4 are CG-2345 CH-1234 may expand MCLQ(C)s CEF-12346 CE-123456 (check in this order) CG-2345 expands CE-123456 to CEG-2345 and CH-1234 expands CEF-12346 to CEFH-1234) B C D E F G H I 1 A B C D E F H I 1 G The Ct(C&X)=2 is CI-13 may expand MCLQ(C)s CEFH-1234 CEG-2345 CEF-12346 CE-123456 (check in this order). CI-13 expands CEFH-1234 to CEFHI-13 C D E F H I 1 2 B C D E F H 1 2 G A B C E F H 1 2 D G C D E F H 1 3 C E F H I 1 2 B C E F H 1 3 A C E F H 1 3 C D E G 1 3 C E F H 1 4 D C E 1 4 E C G 1 4 E C F 1 5 E C 1 6 DMCLQ(D) unless Ct(D&X)=Ct(D). 4=Ct(D)=Ct(CD) =Ct(DE) so CDE-1345MCLQ(D). Next Ct=Ct(D)-1=3. 3# CLQ(D)s: DF-134 DG-345 DH-134, so we have DFH-134 and DG-345 Each expands a maximal: DFH-134 expands CDE-1345 to CDEFH-134. DG-345 expands CDE-1345 to CDEG-345 The Ct(D&X)=2 is DI-13 which expands CDEFH-134 to CDEFHI-13

Depth First Bipartite Max Clique Mine on G9 Find all MCLQs containing A: The ApTree is a MCLQ unless there are pairwise ANDs with same count. B A 1 2 C A 1 3 D A 1 2 E A 1 3 F A 1 3 G A 1 2 H A 1 3 A I 1 A J A K A L A M A N C B 1 3 D 2 E F G H I A J K L N M E C 1 6 F 5 G 4 H I 2 L C D 1 4 E F 3 G H I 2 J K L F E 1 6 G H 7 I 3 E F 1 6 G 4 H 7 I H G 1 8 I 5 G H 1 8 I 9 H I 1 9 L J 1 5 C E F G 3 H 4 I K C D E F 1 G 2 H I 3 J L L C E F 1 G 4 H 5 I G M 1 2 I 3 J L A-F N K H G N 1 2 I 3 J L A-F M K H A 1 3 B 1 3 C 1 6 D 1 4 E 1 8 F 1 8 G 1 a H 1 e I 1 c J 1 5 K 1 4 L 1 6 M 1 3 N 1 3 The EpTree is a MCLQ(E) unless there are pairwise ANDs with that EpTree having Ct=Ct(E)=8. None, so E-12345679MCLQ(E). E 1 8 Next check for MCLQs with Ct=Ct(E)-1=7. E1234567 E1234569 E1234579 E1234679 E1235679 E1245679 E1345679 E2345679 are 7 non-Maximal-CLQs. Each Ct(EXpTree)=7=CtE-1, expands one of the 7 into a MCLQ(E). We first check if any pairs give have the same #set by pairwise ANDing them: None give the same #set (There is just one E-AND, EH1234679) H E 1 7 When Ct=1 check the #pTrees containing E (12345679) for maximal using 1st step above. 1 8 2 7 3 4 5 6 9 A B C D E F H I G When Ct=1 check #pTrees containing M(12,13,14). J I L M 1 H N G b 4 c 6 d 7 A B C D E F K Next check for MCLQs with Ct=Ct(M)-1=2. Combine the LetSets of those with the same NumSet. Then combine their LetSets with the LetSet of the , Ct(MG)=Ct(MH)=2 are non-Maximal-CLQs, because they expand IJLMN(12,13,14) to GIJLMN(13,14) and HIJLMN(12,13). H I J L M N 1 2 G I J L M N 1 2 The MpTree is a MCLQ(M) unless there are pairwise M-ANDs with it having Ct=Ct(M)=3. Then Ct(MI)=Ct(MJ)=Ct)ML)=Ct(MN)=3. I J L M N 1 3 Each Ct(EXpTree)=6=Ct(E)-2 (EF, EG) expands one. 1st ANDs, EF&EG= EFG-2347 G E 1 6 F 4 …Each Ct(EXpTree)=3=Ct(E)-5 (i.e., EI) expands one into a MCLQ(E). I E 1 3

The cTrees on G9 B A 1 2 C A 1 3 D A 1 2 E A 1 3 F A 1 3 G A 1 2 H A 1 3 A I 1 A B 1 2 C B 1 3 D B 1 2 E B 1 3 F B 1 3 G B 1 2 H B 1 3 I B 1 2 E C 1 6 F C 1 5 G C 1 4 H C 1 4 I C 1 2 C D 1 4 E D 1 4 F D 1 3 G D 1 3 H D 1 3 I D 1 2 F E 1 6 G E 1 6 H E 1 7 I E 1 3 E F 1 6 G F 1 4 H F 1 7 I F 1 4 H G 1 8 I G 1 5 G H 1 8 I H 1 9 H I 1 9 J F 1 G J 1 3 H J 1 4 L J 1 5 K F 1 G K 1 2 K H 1 I K 1 3 J K 1 2 L K 1 2 L F 1 G L 1 4 H L 1 5 I L 1 5 G M 1 2 H M 1 2 I M 1 3 J M 1 3 M K 1 L M 1 3 N M 1 3 G N 1 2 H N 1 2 I N 1 3 J N 1 3 N K 1 L N 1 3 M N 1 3 A 1 3 B 1 3 C 1 6 D 1 4 E 1 8 F 1 8 G 1 a H 1 e I 1 c J 1 5 K 1 4 L 1 6 M 1 3 N 1 3 1 2 3 4 5 6 7 8 9 a b v d e f g h i For bipartite graphs, define clique Trees (cTrees). The cTrees would have the NUMBERpTree on top and the LETTERpTree on the bottom: B A C D E F G H I J K L M N 1 2 3 4 5 6 7 8 9 a b v d e f g h i 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 C# CL What can we do with cliqueTrees? The cTreeSet of a bipartite graph is closed under AND (i.e., ANDing 2 cTrees is a cTree). The OR of two cTrees with a substantial AND might reveal interesting communities. The cTreeSet of a bipartite graph is closed under OR-AND in which we OR the PART1pTrees and AND the PART2pTrees. In fact, it is exactly that operation we have been doing in all the Bipartite Max Clique Mining algorithms (the AB-LETTERpTree is the LETTER-PART of the OR-AND (OA) of the A-cTree with the B-cTree). On the next slide we start with the “single letter” 1-many cliques, express each as a cTree, then show the AO results on those.

cTree groups on G9 - 2 We have 32 base cliques (1-many and many-1) and three operations which always produce cliques, AND-AND, AND-OR and OR-AND. AND-OR (ao) and OR-AND (oa) are primary operations (AND-AND on these base cliques produces an empty clique (no vertices from the LETTERpart and so no edges) . A 1 3 B 1 3 C 1 6 D 1 4 E 1 8 F 1 8 G 1 a H 1 e I 1 c J 1 5 K 1 4 L 1 6 M 1 3 N 1 3 1 8 2 1 7 3 1 8 4 1 7 5 1 4 6 1 4 7 1 4 8 1 3 9 1 4 a 1 4 b 1 4 c 1 6 d 1 7 e 1 8 f 1 5 g 1 2 h 1 2 i 1 2 B A C D E F G H I J K L M N 1 2 3 4 5 6 7 8 9 a b v d e f g h i 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 C# CL AND-OR (ao) and OR-AND (oa) are symmetric so lets focus on OA. Including the empty clique of pure0 on top and pure1 below (01), (CLQ, oa) has identity, 01, so is a monoid (lacks inverses) Note, (CLQ,oo) is not a monoid since the operation, oo, applied to two cliques does not necessarily (ever?) produce a clique. (CLQ, oa) is commutative. QUESTIONS? 1. Is the BaseCliqueSet (BCS) a basis for (CLQ,oa), i.e. can you get any clique from the base cliques using oa over and over? Yes. If ABCE1456 is a clique, we get it as AoaBoaCoaE because unless A,B,C,E were all connected to 1,4,5,6 as base cliques, then ABCE1456 would not be a clique (be missing edges). So interestingly, (CLQ,oa)=(CLQ,ao), i.e., each of the operations produce all of CLQ from BCS, so we never need both?, although there may be speed reasons to use both. We have 3 commutative monoids (CLQ,oa), (CLQ,ao), (CLQ,aa). 2. Can we find MCLQ using this structure? Do any pairs of these monoids form a group?, ring?, module?, field? E.G., There is a lot of theory developed for finite fields. Wikipedia: A finite (Galois) field is a set on which multiplication, addition, subtraction and division satisfy certain basic rules (e.g., integers mod n , n is prime). A finite field of order q exists iff the order q is a prime power pk. All fields of a given order are isomorphic. In a field of order pk, adding p copies of any element always results in zero; i.e., the characteristic of the field is p. In a finite field of order q, the polynomial Xq − X has all q elements of the finite field as roots. The non-zero elements of a finite field form a multiplicative group. This group is cyclic, so all non-zero elements can be expressed as powers of a single element called a primitive element of the field (there may be several primitive elements for a given field). A field has, by definition, a commutative multiplication operation. A more general algebraic structure that satisfies all the other axioms of a field but isn't required to have a commutative multiplication is called a division ring (or sometimes skewfield). A finite division ring is a finite field by Wedderburn's thm. Finite fields are used in, e.g., number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Finite fields are in class inclusions: Commutative rings⊃ integral domains⊃ integrally closed domains⊃ unique factorization domains⊃ principal ideal domains⊃ Euclidean domains⊃ fields⊃ finite fields Unfortunately CLQ a not group under any of the 3 different operations (oa,ao,aa) since (0,1) under & or | do not have inverses so do not form a group. They are monoid. CLQ is a partially ordered set (POSet) under subgraph inclusion (which is also sub-cTree inclusion: AB iff every 1-position of A is a 1-position of B). A clique in CLQ is maximal (in MCLQ) if it is maximal under this partial order.

MBCLQ(X) X =? 124-ACEFH A B C D E F G H I J K L M N 124-ACEFH 123-BCEFH 12-ABCEFH 123456-CE 1345-CDE 14-ACDEFH 12345679-E (1234678,14)-F (234579,10,13,14,15)-G 24-ACEFGH (12346789.10,11,12,13,15,16)-H (1389.10,11,12,13,14,16,17,18)-I 1-ACEFHI (11,12,13,14,15)-JL (14.15.17.18)-K (10,11,12,13,14,15)-L (12,13,14)-MIJLN Maximal Base CLQs in green (MBCLQs). Theorems? Every MCLQ is generated, using oa, from the base CLQs, YES! 2. Every MCLQ is generated, using oa, from the maximized base CLQs (1st round only). I THINK SO! 3. oa applied to two MBCLQs gives a MCLQ. I HOPE SO! But it looks like it isn’t true. Counterexample Still searching for an efficient way to find MCLQ(A)! There doesn’t seem to be an algebraic method. Probably it will have to be a POSET method? MBCLQ(X) X =? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17,18 1-ABCDEFHI 2-ABCEFGH 3-BCDEFGHI 4-ACDEFGH 345-CDEG 12346-CEFH 2347-EFGH 78-EFG 359-EGHI (10,13)-GHIL (11,12,13)-HIJL (12,13)-HIJLMN 13-GHIJLMN 14-FGIJKLMN 15-GHJKL (1,3,8,9,10,11,12,13,16)-HI (14,17,18)-IK

APPENDIX Breadth 1st Bipartite Clique Thm on G9 (LETpTrees; exhaustive search; elim if Ct=0|1 AAC; BBC; CCE; DCD; MIM; NIN; B A 1 2 C A 1 3 D A 1 2 E A 1 3 F A 1 3 G A 1 2 H A 1 3 C B 1 3 D B 1 2 E B 1 3 F B 1 3 G B 1 2 H B 1 3 I B 1 2 D C 1 4 E C 1 6 F C 1 5 G C 1 4 H C 1 4 I C 1 2 E D 1 4 F D 1 3 G D 1 3 H D 1 3 I D 1 2 F E 1 6 G E 1 6 H E 1 7 I E 1 3 G F 1 4 H F 1 7 I F 1 4 H G 1 8 I G 1 5 J G 1 3 K G 1 2 L G 1 4 M G 1 2 N G 1 2 I H 1 9 J H 1 4 L H 1 5 M H 1 2 N H 1 2 J I 1 4 K I 1 3 L I 1 5 M I 1 3 N I 1 3 K J 1 2 L J 1 2 L K 1 2 M L 1 3 N L 1 3 N M 1 3 A 1 3 B 1 3 C 1 6 D 1 4 E 1 8 F 1 8 G 1 a H 1 e I 1 c J 1 5 K 1 4 L 1 6 M 1 3 N 1 3 Cliques: ABCEFH12; ADEFH13; BDEFHI13; GIJLMN(13,14); HLMN(12,13); ILMN(12,13,14); B A C 1 2 B A E 1 2 B A F 1 2 B A H 1 2 C A D 1 2 C A E 1 3 C A F 1 3 C A G 1 2 C A H 1 3 D A E 1 2 D A F 1 2 D A H 1 2 E A F 1 3 E A G 1 2 E A H 1 3 F A G 1 2 F A H 1 3 G A H 1 2 C B D 1 2 C B E 1 3 C B F 1 3 C B G 1 2 C B H 1 3 C B I 1 2 D B E 1 2 D B F 1 2 D B H 1 2 D B I 1 2 E B F 1 3 E B G 1 2 E B H 1 3 E B I 1 2 F B G 1 2 F B H 1 3 F B I 1 2 G B H 1 2 H B I 1 2 D C E 1 4 D C F 1 3 D C G 1 3 D C H 1 3 D C I 1 2 E C F 1 5 E C G 1 4 E C H 1 4 E C I 1 2 F C G 1 3 F C H 1 4 F C I 1 2 G C H 1 3 H C I 1 2 E D F 1 3 E D G 1 3 E D H 1 3 E D I 1 2 F D G 1 2 F D H 1 3 F D I 1 2 G D H 1 2 H D I 1 2 F E G 1 4 F E H 1 6 F E I 1 2 G E H 1 5 G E I 1 2 H E I 1 3 G F H 1 4 H F I 1 3 A B C E F H 1 2 G I J L M N 1 2 A D E F H 1 2 B D E F H 1 2 A C D E 1 2 A C D F 1 2 A C D H 1 2 A C E F 1 3 A C E G 1 2 A C E H 1 3 A C F G 1 2 A C F H 1 3 A C G H 1 2 I L M N 1 2 H G I 1 4 H G J 1 2 H G L 1 3 I G J 1 2 I G L 1 2 I G M 1 2 I G N 1 2 J G K 1 2 J G L 1 3 J G M 1 2 J G N 1 2 K G L 1 2 L G M 1 2 L G N 1 2 M G N 1 2 I H J 1 3 I H L 1 4 I H M 1 2 I H N 1 2 J H L 1 4 J H M 1 2 J H N 1 2 L H M 1 2 L H N 1 2 M H N 1 2 J I L 1 5 J I M 1 3 J I N 1 3 L I M 1 3 L I N 1 3 M I N 1 3 K J L 1 2 M L N 1 3

G9 Bipartite Clique Thm on G9 (#pTrees) H I F E C K A L D M J B G N Maximal Cliques containing 1 2 1 3 5 4 6 7 8 9 a b c e d f g h i 1 2 6 3 7 4 5 8 9 a b c d e f g h i W 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 8 B A C D E F G H I J K L M N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 B A C D E F G H I J K L M N B A C D E F G H I J K L M N 8 7 8 8 4 4 4 3 4 4 4 6 7 8 5 2 2 2 Next I x out those that are contained in the next level (or have Ct=0 which means there is no subgraph). Each WpTree is itself a 1-many clique, maximal iff no other contains it, Only 3 4 14 qualify based on Ct but none contains (“& with 1” count<8). Many of these3-many are contained in uncomputed also. 13a,13b,13c,13d,13g13abcde and 13h,13i13hi Each 2W pTree is a 2-many clique, maximal iff no 3-many contains it (check later) 2 1 3 4 5 6 7 8 9 a b c e d f g There are 2-many cliques that are contained in 3-many which weren’t computed, such as 15135, 18 138, 19139, 1a13a, 1b13b, 1c13c, 1e13e, 1d13d, 1g13g, 1h13h, 1i13i. 2 1 3 4 5 6 7 8 9 a b c e d f g G9 A B C D E F G H I J K L M N Bipartite graph of the Southern Women Event Participation. Women are numbers, events are letters. B A C D E F G H I J K L M N 1 2 3 4 5 6 7 9 1 3 a b c d e 2 1 2 3 4 5 6 8 9 a b c e d f g 7 3 1 h i 1 3 5 1 3 8 1 3 9 B A C D E F G H I J K L M N B A C D E F G H I J K L M N If numbers=investors and letters=stock (recommends relationship), would we like a clique with many investors and many stocks? eg, MaxClique 12346CEFH (5 investors, 4 stocks). Is 12345679E (8 investor, 1 stock) better? A K-1 clique  an original pTree (e.g., 12345679E  LETTERpTree (E) (actually =). Thus, we can remove if Ct= 0 or 1.

Bipartite Clique Thm on G9 (LETpTrees; &ing w highest Ct; elim if Ct=0|1) I H G B 1 C D E 2 F J L M N I H A 1 B 2 C D E 3 F G 4 J L M N A H 1 3 B C 5 D E 7 F G 8 I 9 J 4 K L M 2 N A 1 3 B 1 3 C 1 6 D 1 4 E 1 8 F 1 8 G 1 a H 1 e I 1 c J 1 5 K 1 4 L 1 6 M 1 3 N 1 3 1 2 3 4 5 6 7 8 9 a b c d e f g h i Using LETpTrees; & w lowest Ct; elim Ct=0|1) 6-2 CLQ: ABCEFH12 Of course we should do an exhaustive search! B A C 1 2 D E F G H B A 1 2 C 3 D E 4 F G H I J L M N K 5 6 7 8 9 a b c d e f g h i A B C D E F G H I J K L M N G9 Bipartite graph of Southern Women Event Participation. Women=#s, events=letters. Or a recommender: Analyst=#s, stocks=letters

Depth First Bipartite Max Clique Mine on G9 Do all pairwise ANDs (including with itself). Working down wrt Cts, AND with all MCLQs with Ct for containment. (Ct=CtX, containment is automatic) B A 1 2 C A 1 3 D A 1 2 E A 1 3 F A 1 3 G A 1 2 H A 1 3 A I 1 A J A K A L A M A N C B 1 3 D 2 E F G H I A J K L N M E C 1 6 F 5 G 4 H I 2 L C D 1 4 E F 3 G H I 2 J K L F E 1 6 G H 7 I 3 E F 1 6 G 4 H 7 I H G 1 8 I 5 G H 1 8 I 9 H I 1 9 L J 1 5 C E F G 3 H 4 I K C D E F 1 G 2 H I 3 J L L C E F 1 G 4 H 5 I G M 1 2 I 3 J L A-F N K H G N 1 2 I 3 J L A-F M K H A 1 3 B 1 3 C 1 6 D 1 4 E 1 8 F 1 8 G 1 a H 1 e I 1 c J 1 5 K 1 4 L 1 6 M 1 3 N 1 3 A B C D E F H I 1 B C D E F G H I 1 A B C E F G H 1 A C D E F G H 1 D C E G 1 E C F H 1 C E F H I 1 2 C E F H 1 4 E C G 1 4 E C F 1 5 E C 1 6

The cTrees on G9 A 1 3 B A 1 2 C A 1 3 D A 1 2 E A 1 3 F A 1 3 G A 1 2 3 B A 1 2 C A 1 3 D A 1 2 E A 1 3 F A 1 3 G A 1 2 H A 1 3 A I 1 A B 1 2 C B 1 3 D B 1 2 E B 1 3 F B 1 3 G B 1 2 H B 1 3 I B 1 2 E C 1 6 F C 1 5 G C 1 4 H C 1 4 I C 1 2 C D 1 4 E D 1 4 F D 1 3 G D 1 3 H D 1 3 I D 1 2 F E 1 6 G E 1 6 H E 1 7 I E 1 3 E F 1 6 G F 1 4 H F 1 7 I F 1 4 H G 1 8 I G 1 5 G H 1 8 I H 1 9 H I 1 9 J F 1 G J 1 3 H J 1 4 L J 1 5 K F 1 G K 1 2 K H 1 I K 1 3 J K 1 2 L K 1 2 L F 1 G L 1 4 H L 1 5 I L 1 5 G M 1 2 H M 1 2 I M 1 3 J M 1 3 M K 1 L M 1 3 N M 1 3 G N 1 2 H N 1 2 I N 1 3 J N 1 3 N K 1 L N 1 3 M N 1 3 A 1 3 B 1 3 C 1 6 D 1 4 E 1 8 F 1 8 G 1 a H 1 e I 1 c J 1 5 K 1 4 L 1 6 M 1 3 N 1 3 1 2 3 4 5 6 7 8 9 a b v d e f g h i A C E F H 1 2 A C E F H 1 2 A C E F H 1 2 B A 1 2 A D 1 A G 1 A I 1 D A 1 2 A G 1 A I 1 G A 1 2 A I A B C E F H 1 2 A C D E F H 1 2 A C E F G H 1 2