A matrix is called nonnegative if all its entries are nonnegative. A matrix \(A\) is called positive semidefinite (resp., positive definite) if \(A\) is symmetric and \(\langle x, Ax\rangle \geq 0\) for all \(x \in \mathbb{R}^n\) (resp., \(\langle x, Ax\rangle > 0\) for all \(x \in \mathbb{R}^n\)).
Let \(\mathbb{P}_G([0, \infty))\) and \(\mathbb{P}_G'([0, \infty))\) be the sets of positive semidefinite and positive definite matrices of order \(n\), respectively, with nonnegative entries, where some positions of zero entries are restricted by a simple graph \(G\) of order \(n\). Given a connected simple graph \(G\) of order \(n \geq 3\), it is shown that the set of powers preserving positive semidefiniteness on \(\mathbb{P}_G([0, \infty))\) is precisely the same as the set of powers preserving positive definiteness on \(\mathbb{P}_G'([0, \infty))\). Based on chain sequences, it is shown that the Hadamard powers preserving the positive (semi) definiteness of every tridiagonal matrix with nonnegative entries are precisely \(r \geq 1\). The infinite divisibility of tridiagonal matrices is studied. The same results are proved for a special family of pentadiagonal matrices.
All the results obtained in the paper are interesting.