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

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