Question

Let ?? be the path with ? edges and ?? (?) be the characteristic polynomial of ??; recall that ?? (?) = det(? − ??) where ? is the adjacency matrix of ??. Use properties of determinants of adjacency matrices to show that for ? ≥ 3, ?? (?) = −? ??−1 (?) − ??−2 (?). It is acceptable to examine a generic adjacency matrix ? for a path and expand det(? − ??) along the first row.

Answer #1

Note that for a path, the adjacency matrix has 2n many ones in
symmetric positions. Then by proper labelling, we can say that the
adjacency matrix of the path P_{n} is (n greater than
2)

Therefore,

Hence,

From this, the result follows. (The sign of the determinant in line 3 has changed since a column operation has performed on the 2nd matrix.

true/false
An unweighted path length measures the number of edges in a
graph.
Breadth first search traverses the graph in "layers", beginning
with the closest nodes to the ending location first.
The computer knows about neighbors by checking the graph storage
(such as the adjacency matrix or the adjacency list).
Breadth first traversals use a stack to process nodes.
The weighted path length is the sum of the edge costs on a
path.
Dijkstra's shortest path algorithm can be used...

Let A and B be 3x3 matricies, with det A = 9 and deb B = 6. Use
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-5
4
0
0
-2
3
-1
2
0
(Use x instead of λ.)
p(x)=

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Please, can you send me the answer today!!

Let G = (V,E) be a graph with n vertices and e edges. Show that
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Let z0 be a zero of the polynomial
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1.
(an ̸=0)
(k=2,3,...).
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zk−zk=(z−z)(zk−1+zk−2z +···+zzk−2+zk−1)
00000 (b) Use the factorization in part (a) to show that
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Let B = {(1, 3), (?2, ?2)} and B' = {(?12, 0), (?4, 4)} be bases
for R2, and
let A =
3
2
0
4
be the matrix for T: R2 ? R2 relative to B.
(a) Find the transition matrix P from B' to B. P =
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Let us consider Boruvka/Sollin's algorithm as shown .
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