Pre-trained sentence representations are crucial for identifying significant sentences in unsupervised document extractive summarization. However, the traditional two-step paradigm of pre-training and sentence-ranking, creates a gap due to differing optimization objectives. To address this issue, we argue that utilizing pre-trained embeddings derived from a process specifically designed to optimize informative and distinctive sentence representations helps rank significant sentences. To do so, we propose a novel graph pre-training auto-encoder to obtain sentence embeddings by explicitly modelling intra-sentential distinctive features and inter-sentential cohesive features through sentence-word bipartite graphs. These fine-tuned sentence embeddings are then utilized in a graph-based ranking algorithm for unsupervised summarization. Our method is a plug-and-play pre-trained model that produces predominant performance for unsupervised summarization frameworks by providing summary-worthy sentence representations. It surpasses heavy BERT- or RoBERTa-based sentence representations in downstream tasks.