Effectively representing graph node sequences in the form of vector embeddings is critical to many applications. We achieve this by (i) first learning vector embeddings of single graph nodes and (ii) then composing them to compactly represent node sequences. Specifically, we propose SENSE-S (Semantically Enhanced Node Sequence Embedding-for Single nodes), a skip-gram based novel embedding mechanism, for single graph nodes that co-learns graph structure as well as their textual descriptions. We demonstrate that SENSE-S vectors increase the accuracy of multi-label classification tasks by up to 50% and link-prediction tasks by up to 78% under a variety of scenarios using real datasets. Based on SENSE-S, we next propose generic SENSE to compute composite vectors that represent a sequence of nodes, where preserving the node order is important. We prove that this approach is efficient in embedding node sequences, and our experiments on real data confirm its high accuracy.