In today's session, we move to the fascinating realm of graph structures. We'll dive deeper into examining one distinctive way of representing graphs - using an Adjacency Matrix. Our specific objective is to unravel how to construct an Adjacency Matrix to depict a given graph using Python, comprehend its underlying structure, and ascertain when this particular representation becomes most advantageous.

To help understand the role of an Adjacency Matrix, let's consider a practical scenario. Suppose you're using social media. Every time you connect with someone — by accepting their friend request, adding them back to your circle, or linking profiles — you're building a real-life graph where nodes represent users’ profiles and edges signify their connections. Now, how would you represent this social network graph so that you can quickly identify who's connected to whom? This is a perfect use case for an Adjacency Matrix representation.

So, what is an Adjacency Matrix? In essence, it is a square matrix, a two-dimensional array, where each cell (i, j) signifies the weight of the edge between vertices i and j in the graph. Distinct from other matrix-like representations, the most striking aspect of an Adjacency Matrix is its ability to provide a concise, easy-to-understand form of visualizing and depicting the vertex connections in any given graph.

To illustrate, let's consider a simple scenario of a small group of friends: Alice, Bob, and Carol. Let's say Alice is friends with both Bob and Carol, but Bob and Carol don't know each other. In social media terms, Alice would have connections with both Bob and Carol. However, Bob's profile would not show Carol as a connection, and the converse holds true as well. Let's look at this graph: