The general inverse degree and some Hamiltonian properties of graphs

Let \( G = (V, E) \) be a graph with minimum degree at least one. The general inverse degree of \( G \) is defined as \(\sum\limits_{v \in V} \frac{1}{d^{\alpha}(v)}\), where \( \alpha \) is a real number with \( \alpha > 0 \). In this paper, we present sufficient conditions involving the general inverse degree with \( \alpha \geq 1 \) for some Hamiltonian properties of graphs and upper bounds for the general inverse degree with \( \alpha \geq 1 \).