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Let ?? be the path with ? edges and ?? (?) be the characteristic polynomial of...

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.

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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 Pn 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.

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