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In this paper we investigate neural graph embeddings as front-end features for various deep neural network (DNN) architectures for speech recognition. Neural graph embedding features are produced by an autoencoder that maps graph structures defined over speech samples to a continuous vector space. The resulting feature representation is then used to augment the standard acoustic features at the input level of a DNN classifier. We compare two different neural graph embedding methods, one based on a local neighborhood graph encoding, and another based on a global similarity graph encoding. They are evaluated in DNN-HMM-based and LSTM-CTC-based ASR systems on a 110-hour Switchboard conversational speech recognition task. Significant improvements in word error rates are achieved by both methods in the DNN-HMM system, and by global graph embeddings in the LSTM-CTC system.