Real-world networks often exhibit community structures, and community query is a fundamental task in graph data mining. Existing studies introduced various models to identify communities within networks, such as k-core based models and k-truss based models. Nevertheless, these models typically confine themselves to constraining the number of neighbors of nodes or edges within a community, disregarding the relationships between these neighbors, namely, the neighborhood structure of the nodes. Consequently, the localized density of nodes within communities tends to be low. To address this limitation, this study integrates the information regarding the neighborhood structure of nodes into the k-core dense subgraph model, thereby introducing a community model based on neighborhood k-core and defining the density of a community. Based on the novel model, this study investigates the densest single community query problem which outputs the community containing the query node set with the highest community density. In real-life networks, the query nodes may be distributed across multiple disjoint communities. To this end, this study further works on the problem of multi-community query based on a density threshold. This entails returning multiple communities that encompass the query node set, with each community demonstrating a density no lower than the user-specified threshold. For the problem of the densest single community query and the multi-community query based on a density threshold, this study introduces the concept of edge density with which the basic algorithms are proposed. To improve the efficiency, the index tree and the enhanced index tree structures are devised to support outputting results in polynomial time. The effectiveness of the community model based on neighborhood k-core and the efficiency of query algorithms are demonstrated through comparative analyses against basic algorithms using several different datasets.