An Enumeration of Certain Equivalence Classes of Directed Multigraphs Having 0 to 5 Nodes with Each Node Having 2 Incoming and 2 Outgoing Arcs

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

1/14

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


2/14

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


3,3Class: 3,3,3


3,8Class: 3,8,8



3/14

Z-18 Monday, March 22, 1999 3

4 NODES4,5Class: 4,5,96Polynomial:x11x11x22x24x33x34x42x43

4,1Class: 4,1,1Polynomial:x11x11x22x22x33x33x44x44


4,2

Class: 4,2,24Polynomial:x11x11x22x22x33x34x44x43

4,7


4,3Class: 4,3,6

Polynomial:x11x11x22x22x34x34x43x43

4,8Class: 4,8,32


4,4Class: 4,4,64


4,9Class: 4,9,48



4/14






4,12


4,17


4,13Class: 4,13,192


4,18Class: 4,18,96


4,14Class: 4,14,96


4,19Class: 4,19,384



5/14


6/14


5,4Class: 5,4,160Polynomial:x11x11x22x22x33x34x44x45x55x53




5,6

Class: 5,6,240Polynomial:x11x11x22x22x33x34x45x45x53x54

5,11


5,7Class: 5,7,20

Polynomial:x11x11x22x22x34x34x45x45x53x53

5,12Class: 5,12,120


5,8Class: 5,8,80


5,13Class: 5,13,960



7/14






5,16


5,21


5,17Class: 5,17,960


5,22Class: 5,22,30


5,18Class: 5,18,480


5,23Class: 5,23,480



8/14






5,26


5,31


5,27Class: 5,27,768


5,32Class: 5,32,160


5,28Class: 5,28,960


5,33Class: 5,33,1920



9/14






5,36


5,41


5,37Class: 5,37,1920


5,42Class: 5,42,960


5,38Class: 5,38,3840


5,43Class: 5,43,3840



10/14






5,46


5,51


5,47Class: 5,47,1920


5,52Class: 5,52,3840


5,48Class: 5,48,1920


5,53Class: 5,53,1920



11/14






5,56


5,61


5,57Class: 5,57,960


5,62Class: 5,62,480


5,58Class: 5,58,3840


5,63Class: 5,63,1920



12/14






5,66


5,71


5,67Class: 5,67,3840


5,72Class: 5,72,80


5,68Class: 5,68,1920


5,73Class: 5,73,480



13/14






5,76


5,81


5,77Class: 5,77,480


5,82Class: 5,82,384


5,78Class: 5,78,480


5,83Class: 5,83,1920



14/14




Documents

An Enumeration of Certain Equivalence Classes of Directed Multigraphs Having 0 to 5 Nodes with Each Node Having 2 Incoming and 2 Outgoing Arcs