Welcome to our session on Graph Algorithms Implementation. A large proportion of real-world problems, from social networking to routing applications, can be represented by graphs. Understanding and implementing graph algorithms is thus a key skill to have in your programming toolkit. In this lesson, we introduce and explore one of the most fundamental graph traversal algorithms — the Breadth-First Search (BFS).

Let's take a sneak peek at the BFS algorithm. Given a graph and a starting vertex, BFS systematically explores the edges of the graph to "visit" each reachable vertex. It does this by managing a queue of vertices yet to be explored and consistently visiting all vertices adjacent to the current one before moving on. BFS is particularly efficient because it can find the shortest distance between the starting vertex and all other vertices in a graph.

Here's a sample BFS algorithm implemented in PHP that we will dissect in depth and master:

As we delve into this session, we will understand the mechanics behind BFS. Our study will include the concepts of traversal, the queue data structure's usefulness in BFS, and how to handle the discovery and processing of nodes. Equipped with these fundamentals, we'll practice a variety of problems calling for BFS to perform node-level searches or find connected components in a graph. Let's dive in and uncover the power of graph algorithms!