So my course is just getting further and further above my head. I got most of them but I'm stuck on these 3 and have no idea how to start them.
This is due friday so hopefully I can get input before that! :)
1) Let G be a (2k + 1)-regular graph, and assume that G - S is still...
Homework Statement
Characterize all possible sequences d1; d2; : : : ; dn so that there exists a forest
with vertex set fv1; v2; : : : vng with deg(vi) = di. (So, you should nd a statement of the
form: a sequence d1; : : : dn comes from a forest if and only if ... )
I emailed him and...
as for the bipartite part, if we have a bipartite regular graph, then there must be an even number of vertices, n.
If we split there vertices into V1 and V2 where every edge has one endpoint in V1 and one endpoint in V2, then we can label the vertices of V1 1,2,3...n/2 and label V2 with...
just generally. And why is my argument circular?
ax=lambdax is the definition of an eigenvalue.
If i let y=lambdax then ax=y where y is the vector resulting from the product of ax.
I know that k is an eigenvalue.
if x is some vector [x1,x2,..xj,...xi] where xj is the largest of those...
Could I say something like...
Ax=lambda*x for some vector x and the adjacency matrix A, (giving a eigenvalue of lambda).
**this is a true formula**
Let x be a vector and y=lambda*x be the product vector.
Consider the product of A and any vector x. The largest value in the product vector (y)...
oh, so a matrix M multiplied by (1,1,...1) will give a result such that it equals N*(1,1,1...1) where N is an eigenvalue? is that a property?
and then I need to show that this is the largest eigenvalue?
so I think I get that part, not sure how to show its the largest.
I've browsed...
So if I had say, the matrix A
0 1 0 1
1 0 1 0
0 1 0 1
1 0 1 0
Then A X (1,1,1,1) would be 2,2,2,2 correct? and these numbers will always be the value of k?
thanks for the reply. I get that the matrix is symetric and that each row has the same number of 1's.
Where you lost me was
"consider columns & the action of the vector (1,1,...,1) this shows d is an eigenvalue"
1) why the columns?
2) how do you consider the column if you don't know how many...
Homework Statement
Let G be a simple, connected d-regular graph and let A be its adjacency matrix.
(i) Show that A has largest eigenvalue d
(ii) Show that A has an eigenvalue of -d if and only if G is bipartite.
Homework Equations
Don't know?The Attempt at a Solution
I really have no idea...
1) Problem 1 For every i; j with 1<=i; j<=8 fi nd a nim heap which is equivalent to the i x j
game of Chop.
CHOP Game Start with an i x j array of boxes viewed as a plank which
is secured only at the lower left hand corner. Each player is a
pirate, and they alternate turns. On each turn a...