Co-saliency estimation which aims to identify the common salient object regions contained in an image set is an active problem in computer vision. The main challenge for co-saliency estimation problem is how to exploit the salient cues of both intra-image and inter-image simultaneously. In this paper, we first represent intra-image and inter-image as intra-graph and inter-graph respectively and formulate co-saliency estimation as graph nodes labeling. Then, we propose a novel multiple graph learning and convolutional network (M-GLCN) for image co-saliency estimation. M-GLCN conducts graph convolutional learning and labeling on both inter-graph and intra-graph cooperatively and thus can well exploit the salient cues of both intra-image and inter-image simultaneously for co-saliency estimation. Moreover, M-GLCN employs a new graph learning mechanism to learn both inter-graph and intra-graph adaptively. Experimental results on several benchmark datasets demonstrate the effectiveness of M-GLCN on co-saliency estimation task.