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**1 - 2**of**2**### Integer index in trees of diameter 4

, 2014

"... We characterize when a tree of diameter 4 has integer index and we provide examples of infinite families of non-integral trees with integer index. We also determine a tight upper bound for the index of any tree of diameter 4 based on its maximum degree. Moreover, we present a new infinite family o ..."

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We characterize when a tree of diameter 4 has integer index and we provide examples of infinite families of non-integral trees with integer index. We also determine a tight upper bound for the index of any tree of diameter 4 based on its maximum degree. Moreover, we present a new infinite family of integral trees of diameter 4.

### Integral trees with given nullity

"... A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. We prove that for a given nullity more than 1, there are only finitely many integral trees. It is also shown that integral trees with nullity 2 and 3 are unique. ..."

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A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. We prove that for a given nullity more than 1, there are only finitely many integral trees. It is also shown that integral trees with nullity 2 and 3 are unique.