The problem of computing a sparse spanning subgraph is a well-studied problem in the distributed setting, and a lot of research was done in the direction of computing spanners or solving the more relaxed problem of connectivity. Still, efficiently constructing a linear-size spanner deterministically remains a challenging open problem even in specific topologies. In this paper we provide several simple spanner constructions of linear size, for various graph families. Our first result shows that the connectivity problem can be solved deterministically using a linear size spanner within constant running time on graphs with bounded neighborhood independence. This is a very wide family of graphs that includes unit-disk graphs, unit-ball graphs, line graphs, claw-free graphs and many others. Moreover, our algorithm works in the model. It also immediately leads to a constant time deterministic solution for the connectivity problem in the Congested-Clique. Our second result provides a linear size spanner in the model for graphs with bounded diversity. This is a subtype of graphs with bounded neighborhood independence that captures various types of networks, such as wireless networks and social networks. Here too our result has constant running time and is deterministic. Moreover, the latter result has an additional desired property of a small stretch.