Graph Algorithms can be used in many different systems, so you do not need to use Spark or Neo4j to use graph algorithms. We use Spark and Neo4j to showcase graph algorithms in the book because they both had unique qualities. Spark is a popular scale-out computing framework with libraries to support a variety of data science workflows. Neo4j offers a high-performance graph-native platform with over 45 graph algorithms and the ability to persist graphs.
Most people use a graph query when investigating limited areas of a graph or a specific question. For example, how many hops between node A & B. However, we would want to use a graph algorithm when we looking for more holistic analysis of the graph structure. For example, finding all the communities based on the number of relationships among nodes.
If you do not need to store your graph and you're working with a small dataset, there are a number of platforms you might use. NetworkX has quite a few algorithms and is often used in academic settings with small graphs.
We wrote the book so that the concepts about graph analysis and how certain algorithms calculate results could be applied more generally. For those just getting started, you might want to download the free digital copy and check out the first few chapters: https://neo4j.com/graph-algorithms-book/