Rank Matrix
المؤلف:
Biggs, N. L
المصدر:
Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press
الجزء والصفحة:
...
14-4-2022
1932
Rank Matrix
If the rank polynomial 
of a graph 
is given by 
, then 
is the number of subgraphs of 
with rank 
and co-rank 
, and the matrix 
is called the rank matrix of 
.
For example, the rank matrix of the complete bipartite graph 
, which has rank polynomial
 |
(1)
|
is given by
![[1 ; 9 ; 36 ; 84 9 ; 117 45 6 ; 81 78 36 9 1]](https://mathworld.wolfram.com/images/equations/RankMatrix/NumberedEquation2.svg) |
(2)
|
(Biggs 1993, p. 73), and the rank matrix of the Petersen graph is
![[1 ; 15 ; 105 ; 455 ; 1365 12 ; 2991 130 ; 4875 630 30 ; 5805 1725 240 15 ; 4680 2765 816 135 10 ; 2000 2172 1230 445 105 15 1]](https://mathworld.wolfram.com/images/equations/RankMatrix/NumberedEquation3.svg) |
(3)
|
(Godsil and Royle 2001, p. 356).
REFERENCES
Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, p. 73, 1993.
Godsil, C. and Royle, G. Algebraic Graph Theory. New York: Springer-Verlag, 2001.
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