Cell clustering is critical for analyzing single-cell RNA sequencing (scRNA-seq) data. While many deep graph embedding methods have been developed for clustering scRNA-seq data, their usage of graph structures is often coupled with noise and outliers, which may mislead the message passing in graph neural networks (GNNs) and result in suboptimal clustering outcomes. To address this issue, this paper proposes an adaptive graph convolutional network for single-cell RNA-seq data clustering (scAGC). Specifically, scAGC employs a zero-inflated negative binomial (ZINB) model-based autoencoder to capture the global probabilistic structure of the highly sparse and over-dispersed scRNA-seq data. Meanwhile, the adaptive graph convolutional (AGC) module refines the initial graph topology and improves the data representation fidelity through an iterative optimization process. Furthermore, an attention based representation fusion (ABRF) module weights and fuses the data representations from the two aforementioned modules, which contributes to the construction of an improved graph structure and mitigates the over-smoothing issue in GCNs. Finally, we design a self-adaptive learning (SAL) module optimize the overall network architecture in a self-supervised manner. The experiments conducted on fourteen real-world scRNA-seq datasets demonstrate the superior clustering performance of scAGC over the state-of-the-art.