Some New Sufficient Conditions for a Graph to be $l$-Deficient

For a (molecular) graph $G$ and any real number $\alpha\ne 0$ , the zero-order general Randi\'c index , denote by $^0R_\alpha$, is defined by the following equation: \begin{align*} {^0R_\alpha} (G) =\sum_{v\in G}d_G (v) ^{\alpha} (\alpha \in \mathbb{R}-\left\{0\right\}) . \end{align*} The deficiency of $G$, denoted by $def(G)$, is equal to the cardinality of vertices which are not covered by a maximum matching in $G$. A graph G is called $l$-deficient if $def(G)\le l$. In this paper, we use this index to give sufficient conditions for a connected graph, bipartite graph and a balanced bipartite graph $G$ to satisfy the $l$-deficient property, and show that none of these conditions can be dropped. We will also use these results to enhance and generalise the results that already obtained by M. An and K. C. Das in 2018 and G. Su et al. in 2022.