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8/11/2019 An Enumeration of Certain Equivalence Classes of Directed Multigraphs Having 0 to 5 Nodes with Each Node Having 2 Incoming and 2 Outgoing Arcs
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AN ENUMERATION OF CERTAIN EQUIVALENCE CLASSES OF DIRECTED MULTIGRAPHS HAVING 0 TO 5 NODESWITH EACH NODE HAVING 2 INCOMING AND 2 OUTGOING ARCS
C. C. BriggsCenter for Academic Computing, Penn State University, University Park, PA 16802
March 22,1999
Abstract. An enumeration is given of certain equivalence classes of directed multigraphs having 0 to 5 nodes with each node having 2 incoming and 2 outgoing arcs.
PACS Numbers: 02.10.Eb
1 NODE
1,1Class: 1,1,1Polynomial:x11x11
2 NODES
This paper presents an enumeration of certain equivalence
classes of directed multigraphs having 0 to 5 nodes with each
node having 2 incoming and 2 outgoing arcs.
The classeswhich consist of distinct but isomorphic
multigraphsare (1) designated by sets of 3 numbers, the 1st,
2nd, and 3rd of which specify the numbers of nodes of the
component multigraphs, the ranks, and the cardinalities of the
classes, respectively, and (2) typified by representative
multigraphs characterized by pictures and by corresponding
polynomials in variables individually signifying the
occurrences of directed arcs, each variable having 2
subscripts, the 1st and 2nd of which specify the nodes from
which and at which an arc emanates and terminates,
respectively. (For 0 nodes, the representative multigraph is
the null multigraph, the picture for which is intentionally left
blank and the polynomial taken to be equal to 1.)
For a check, note that, for p nodes, the cardinalities of the
classes add up to (2p)!/2p.
Some numerical features of the classes appear in Table 1
(see below).
TABLE 1. SOME NUMERICAL FEATURES OF THE CLASSES
NUMBER OF NUMBER OF SUM OFNODES CLASSES CARDINALITIES
0 1 1
1 1 12 3 63 8 904 25 25205 85 113,400
0 NODES
2,1
Class: 2,1,1Polynomial:x11x11x22x22
0,1Class: 0,1,1
Polynomial: 1
2,2Class: 2,2,4
Polynomial:x11x12x22x21
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8/11/2019 An Enumeration of Certain Equivalence Classes of Directed Multigraphs Having 0 to 5 Nodes with Each Node Having 2 Incoming and 2 Outgoing Arcs
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2 Monday, March 22, 1999 Z-18
2,3Class: 2,3,1Polynomial:x12x12x21x21
3,4Class: 3,4,16Polynomial:x11x12x22x23x33x31
3 NODES 3,5Class: 3,5,24Polynomial:x11x13x22x23x31x32
3,1
Class: 3,1,1Polynomial:x11x11x22x22x33x33
3,6
Class: 3,6,24Polynomial:x11x12x23x23x31x32
3,2Class: 3,2,12
Polynomial:x11x11x22x23x33x32
3,7Class: 3,7,2
Polynomial:x12x12x23x23x31x31
3,3Class: 3,3,3
Polynomial:x11x11x23x23x32x32
3,8Class: 3,8,8
Polynomial:x12x13x21x23x31x32
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8/11/2019 An Enumeration of Certain Equivalence Classes of Directed Multigraphs Having 0 to 5 Nodes with Each Node Having 2 Incoming and 2 Outgoing Arcs
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Z-18 Monday, March 22, 1999 3
4 NODES4,5Class: 4,5,96Polynomial:x11x11x22x24x33x34x42x43
4,1Class: 4,1,1Polynomial:x11x11x22x22x33x33x44x44
4,6Class: 4,6,96Polynomial:x11x11x22x23x34x34x42x43
4,2
Class: 4,2,24Polynomial:x11x11x22x22x33x34x44x43
4,7
Class: 4,7,8Polynomial:x11x11x23x23x34x34x42x42
4,3Class: 4,3,6
Polynomial:x11x11x22x22x34x34x43x43
4,8Class: 4,8,32
Polynomial:x11x11x23x24x32x34x42x43
4,4Class: 4,4,64
Polynomial:x11x11x22x23x33x34x44x42
4,9Class: 4,9,48
Polynomial:x11x12x22x21x33x34x44x43
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4 Monday, March 22, 1999 Z-18
4,10Class: 4,10,96Polynomial:x11x12x22x23x33x34x44x41
4,15Class: 4,15,192Polynomial:x11x13x22x24x31x34x42x43
4,11Class: 4,11,384Polynomial:x11x12x22x24x33x34x41x43
4,16Class: 4,16,192Polynomial:x11x13x22x24x32x34x41x43
4,12
Class: 4,12,24Polynomial:x11x12x22x21x34x34x43x43
4,17
Class: 4,17,192Polynomial:x11x12x21x23x34x34x42x43
4,13Class: 4,13,192
Polynomial:x11x12x22x23x34x34x41x43
4,18Class: 4,18,96
Polynomial:x11x12x23x23x34x34x41x42
4,14Class: 4,14,96
Polynomial:x11x13x22x23x34x34x41x42
4,19Class: 4,19,384
Polynomial:x11x12x23x24x31x34x42x43
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8/11/2019 An Enumeration of Certain Equivalence Classes of Directed Multigraphs Having 0 to 5 Nodes with Each Node Having 2 Incoming and 2 Outgoing Arcs
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8/11/2019 An Enumeration of Certain Equivalence Classes of Directed Multigraphs Having 0 to 5 Nodes with Each Node Having 2 Incoming and 2 Outgoing Arcs
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6 Monday, March 22, 1999 Z-18
5,4Class: 5,4,160Polynomial:x11x11x22x22x33x34x44x45x55x53
5,9Class: 5,9,240Polynomial:x11x11x22x23x33x32x44x45x55x54
5,5Class: 5,5,240Polynomial:x11x11x22x22x33x35x44x45x53x54
5,10Class: 5,10,480Polynomial:x11x11x22x23x33x34x44x45x55x52
5,6
Class: 5,6,240Polynomial:x11x11x22x22x33x34x45x45x53x54
5,11
Class: 5,11,1920Polynomial:x11x11x22x23x33x35x44x45x52x54
5,7Class: 5,7,20
Polynomial:x11x11x22x22x34x34x45x45x53x53
5,12Class: 5,12,120
Polynomial:x11x11x22x23x33x32x45x45x54x54
5,8Class: 5,8,80
Polynomial:x11x11x22x22x34x35x43x45x53x54
5,13Class: 5,13,960
Polynomial:x11x11x22x23x33x34x45x45x52x54
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Z-18 Monday, March 22, 1999 7
5,14Class: 5,14,480Polynomial:x11x11x22x24x33x34x45x45x52x53
5,19Class: 5,19,1920Polynomial:x11x11x22x23x34x35x42x45x53x54
5,15Class: 5,15,960Polynomial:x11x11x22x24x33x35x42x45x53x54
5,20Class: 5,20,15Polynomial:x11x11x23x23x32x32x45x45x54x54
5,16
Class: 5,16,960Polynomial:x11x11x22x24x33x35x43x45x52x54
5,21
Class: 5,21,240Polynomial:x11x11x23x23x32x34x45x45x52x54
5,17Class: 5,17,960
Polynomial:x11x11x22x23x32x34x45x45x53x54
5,22Class: 5,22,30
Polynomial:x11x11x23x23x34x34x45x45x52x52
5,18Class: 5,18,480
Polynomial:x11x11x22x23x34x34x45x45x52x53
5,23Class: 5,23,480
Polynomial:x11x11x23x23x34x35x42x45x52x54
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8/11/2019 An Enumeration of Certain Equivalence Classes of Directed Multigraphs Having 0 to 5 Nodes with Each Node Having 2 Incoming and 2 Outgoing Arcs
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8 Monday, March 22, 1999 Z-18
5,24Class: 5,24,240Polynomial:x11x11x23x24x32x35x42x45x53x54
5,29Class: 5,29,3840Polynomial:x11x12x22x23x33x35x44x45x51x54
5,25Class: 5,25,480Polynomial:x11x11x23x24x32x35x43x45x52x54
5,30Class: 5,30,1920Polynomial:x11x12x22x25x33x34x44x45x51x53
5,26
Class: 5,26,640Polynomial:x11x12x22x21x33x34x44x45x55x53
5,31
Class: 5,31,960Polynomial:x11x12x22x21x33x34x45x45x53x54
5,27Class: 5,27,768
Polynomial:x11x12x22x23x33x34x44x45x55x51
5,32Class: 5,32,160
Polynomial:x11x12x22x23x33x31x45x45x54x54
5,28Class: 5,28,960
Polynomial:x11x12x22x21x33x35x44x45x53x54
5,33Class: 5,33,1920
Polynomial:x11x12x22x23x33x34x45x45x51x54
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8/11/2019 An Enumeration of Certain Equivalence Classes of Directed Multigraphs Having 0 to 5 Nodes with Each Node Having 2 Incoming and 2 Outgoing Arcs
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Z-18 Monday, March 22, 1999 9
5,34Class: 5,34,1920Polynomial:x11x12x22x24x33x34x45x45x51x53
5,39Class: 5,39,80Polynomial:x11x12x22x21x34x34x45x45x53x53
5,35Class: 5,35,3840Polynomial:x11x12x22x24x33x35x41x45x53x54
5,40Class: 5,40,320Polynomial:x11x12x22x21x34x35x43x45x53x54
5,36
Class: 5,36,3840Polynomial:x11x12x22x24x33x35x43x45x51x54
5,41
Class: 5,41,1920Polynomial:x11x12x22x23x31x34x45x45x53x54
5,37Class: 5,37,1920
Polynomial:x11x14x22x25x33x34x42x45x51x53
5,42Class: 5,42,960
Polynomial:x11x12x22x23x34x34x45x45x51x53
5,38Class: 5,38,3840
Polynomial:x11x14x22x25x33x35x41x42x53x54
5,43Class: 5,43,3840
Polynomial:x11x12x22x23x34x35x41x45x53x54
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10 Monday, March 22, 1999 Z-18
5,44Class: 5,44,240Polynomial:x11x13x22x23x31x32x45x45x54x54
5,49Class: 5,49,1920Polynomial:x11x13x22x24x31x34x45x45x52x53
5,45Class: 5,45,1920Polynomial:x11x13x22x23x31x34x45x45x52x54
5,50Class: 5,50,3840Polynomial:x11x13x22x25x31x34x42x45x53x54
5,46
Class: 5,46,480Polynomial:x11x13x22x23x34x34x45x45x51x52
5,51
Class: 5,51,1920Polynomial:x11x13x22x25x31x34x43x45x52x54
5,47Class: 5,47,1920
Polynomial:x11x13x22x23x34x35x41x45x52x54
5,52Class: 5,52,3840
Polynomial:x11x13x22x25x32x34x41x45x53x54
5,48Class: 5,48,1920
Polynomial:x11x13x22x24x31x32x45x45x53x54
5,53Class: 5,53,1920
Polynomial:x11x13x22x25x32x34x43x45x51x54
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8/11/2019 An Enumeration of Certain Equivalence Classes of Directed Multigraphs Having 0 to 5 Nodes with Each Node Having 2 Incoming and 2 Outgoing Arcs
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Z-18 Monday, March 22, 1999 11
5,54Class: 5,54,1920Polynomial:x11x13x22x25x34x34x42x45x51x53
5,59Class: 5,59,240Polynomial:x11x12x23x23x31x32x45x45x54x54
5,55Class: 5,55,1920Polynomial:x11x13x22x25x34x35x41x42x53x54
5,60Class: 5,60,960Polynomial:x11x12x23x23x31x34x45x45x52x54
5,56
Class: 5,56,1920Polynomial:x11x12x21x23x32x34x45x45x53x54
5,61
Class: 5,61,960Polynomial:x11x12x23x23x32x34x45x45x51x54
5,57Class: 5,57,960
Polynomial:x11x12x21x23x34x34x45x45x52x53
5,62Class: 5,62,480
Polynomial:x11x12x23x23x34x34x45x45x51x52
5,58Class: 5,58,3840
Polynomial:x11x12x21x23x34x35x42x45x53x54
5,63Class: 5,63,1920
Polynomial:x11x12x23x23x34x35x41x45x52x54
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12 Monday, March 22, 1999 Z-18
5,64Class: 5,64,1920Polynomial:x11x12x23x24x31x32x45x45x53x54
5,69Class: 5,69,3840Polynomial:x11x12x23x24x32x35x43x45x51x54
5,65Class: 5,65,1920Polynomial:x11x12x23x24x31x34x45x45x52x53
5,70Class: 5,70,1920Polynomial:x11x12x23x24x34x35x43x45x51x52
5,66
Class: 5,66,3840Polynomial:x11x12x23x24x31x35x42x45x53x54
5,71
Class: 5,71,20Polynomial:x12x12x21x21x34x34x45x45x53x53
5,67Class: 5,67,3840
Polynomial:x11x12x23x24x31x35x43x45x52x54
5,72Class: 5,72,80
Polynomial:x12x12x21x21x34x35x43x45x53x54
5,68Class: 5,68,1920
Polynomial:x11x12x23x24x32x34x45x45x51x53
5,73Class: 5,73,480
Polynomial:x12x12x21x23x31x34x45x45x53x54
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Z-18 Monday, March 22, 1999 13
5,74Class: 5,74,480Polynomial:x12x12x21x23x34x34x45x45x51x53
5,79Class: 5,79,960Polynomial:x12x12x23x24x31x35x41x45x53x54
5,75Class: 5,75,1920Polynomial:x12x12x21x23x34x35x41x45x53x54
5,80Class: 5,80,1920Polynomial:x12x12x23x24x31x35x43x45x51x54
5,76
Class: 5,76,24Polynomial:x12x12x23x23x34x34x45x45x51x51
5,81
Class: 5,81,1920Polynomial:x12x13x21x23x34x35x41x45x52x54
5,77Class: 5,77,480
Polynomial:x12x12x23x23x34x35x41x45x51x54
5,82Class: 5,82,384
Polynomial:x12x13x21x24x31x35x42x45x53x54
5,78Class: 5,78,480
Polynomial:x12x12x23x24x31x34x45x45x51x53
5,83Class: 5,83,1920
Polynomial:x12x13x21x24x31x35x43x45x52x54
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14 Monday, March 22, 1999 Z-18
5,84Class: 5,84,1920Polynomial:x12x13x21x24x32x35x43x45x51x54
5,85Class: 5,85,768Polynomial:x12x13x23x24x34x35x41x45x51x52