Path Matrix

When you start learning graphs, you meet many new terms — adjacency matrix, incidence matrix, degree, and so on.
Among them, Path Matrix is one of the simplest ideas, but it often looks confusing at first.
So let’s understand it slowly, gently… like a teacher sitting next to you with a pen and paper.

🌿 What Is a Path Matrix?

Think of a graph as a map of places (vertices) connected by roads (edges).
Now imagine you want a quick answer to one question:

“Is there ANY possible way to travel from one vertex to another?”

You don’t care how long the route is.
You don’t care how many turns it takes.
You just want to know whether a route exists.

That yes/no information is stored in a Path Matrix.

🧠 Simple Definition (in easy words)

A Path Matrix is a table that shows whether a path exists between any two vertices in a graph.

Write 1 if a path exists.
Write 0 if no path exists.

This is like a “reachable map.”

🎯 Important Point:

A vertex is always reachable from itself, so the diagonal entries are typically 1.

🌼 Let’s Start with a Small Example

Consider this graph:

   (A) ---- (B)
      \      |
       \     |
        (C)  |
          \  |
           (D)

Connections:

A connected to B
A connected to C
B connected to D
C connected to D

You can think of it like four friends living in nearby houses with walking paths between them.

🔍 Which vertex can reach which?

Let’s explore reachability in a simple, conversational way:

From A, you can reach B (direct), reach C (direct), and reach D (through B or C).
From B, you can reach A (backtrack), reach D, and reach C (B → A → C).
From C, you can reach A, D, and B (C → A → B).
From D, you can reach B, reach C, and reach A.

This means everybody can reach everybody — maybe not directly, but eventually.

📘 Now Build the Path Matrix

Order of vertices: A, B, C, D

We’ll fill in 1s and 0s based on reachability.

          A   B   C   D
        -----------------
A   |     1   1   1   1
B   |     1   1   1   1
C   |     1   1   1   1
D   |     1   1   1   1

Why all 1s?

Because every vertex can reach every other vertex somehow.

🎨 Simple Diagram of the Path Matrix

Here’s a clean visualization:

Path Matrix P(G):

        A   B   C   D
      -----------------
A  →    1   1   1   1
B  →    1   1   1   1
C  →    1   1   1   1
D  →    1   1   1   1

Imagine this as a “reachable checklist.”
Everything is reachable, so everything is marked as 1.