Dear GAP Forum,
This was my unsolved question:
I want to calculate all eigenvalues of cubic cayley graphs associate
with sporadic simple group. So if G denote one of 26 sporadic simple
group, and X a set of generators with 3 elements, one "a" of order 2,
and the other "b" and "b^-1" of order greater than 2. then Cay(G,X) is
a cubic Cayley graph. and by a well-know formula for computing
eigenvalues of cayley graphs, we have: for each
"Xi" in Irr(G)
of degree n, we have n eigenvalues L_1, L_2,...,L_n each of them has
multiplicity n ( so in total this character "Xi" of
degree n gives us "n^2" eigenvalues of our graph). and we can compute them by
Newton-Waring identities and following fact that
for t from 1 to n we have (L_1)^t + (L_2)^t + ... +(L_n)^t = Sum {Xi
(x_1.x_2....x_t)} where sum is over the value of "Xi" on
the all product of t elemnts from our generator set X.
Using
this I will find a polynomial of degree n and roots give me the
eigenvalues, instead of determaining the characterestic polynomial of a
matrix of order |G| which for sporadic simple group this number is huge
and also finding their roots!
As you see, I need to calculate
this for every character in Irr(G), and evalute them in all t-product
of elements of generator set.
I know this will involve lots of calculation as mentioned it to me by Thomas
Breuer. Because of the special case here, I am just having a generator set of
3 elements, I would like to know is it any function to calculate for generator
"a" and "b" and conjugacy class "C" the number "n_{t,C}(a,b)" which calculates
the number of t-products( products of length t of a,a^-1, b, b^-1) of a and b
which belongs to the Conjugacy class "C" for t runs from 1 to max{ deg(Xi) |
"Xi" runs over Irr(G)}?
Because after this point most of calculations above are pretty simple.
Azhvan
_________________________________________________________________
More storage. Better anti-spam and antivirus protection. Hotmail makes it
simple.
http://go.microsoft.com/?linkid=9671357_______________________________________________
Forum mailing list
Forum@mail.gap-system.org
http://mail.gap-system.org/mailman/listinfo/forum
