Electrical – Certain property of a bus admittance matrix

The clue here is the "open drain".

Basically each output is a switch which is connected to ground. This switch goes between the cathode of your LED and ground, and you feed your power supply into the anode of each LED.

If you look on page 8 of the data sheet you see an example with lots of 7 segment LED displays. You can see the "Vled" is connected to all the displays through a pair of transistors - these are just used to switch between left and right halves of the display to get twice as many digits - and there is no ground connection from the displays. This is because each cathode goes into the chip and then to ground.

The chip itself doesn't source any current, it just sinks it.

As for the current limiting, yes, it is very vague. Looking at the data table you can see that when a segment is off then the output voltage is at 3V.

This is the same voltage you should be providing to the anode of your LEDs.

When the segment is turned on the output voltage "drops" to 1.8V. This means that when the output is "on" it has a voltage drop of 1.8V to ground. With a 3V supply to your LED anode this would equate to 3V - 1.8V = 1.2V for running the LED.

This is the same as having a resistor that drops 1.8v at the current your LED draws.

In your actual circuit, have you connected the wires and the LED leads at the junctions indicated by the black circles, or have you connected them wherever the lines overlap in your schematic diagram?

The correct reading of the schematic is to only have connections where indicated by the junction indicators (black filled circles). If you do so, then the third scenario in your diagram does not exist - the wires going horizontally and vertically do not actually touch at all.

One way to visualize this: The horizontal wires are all on the plane of your work table, the vertical wires are all in a plane floating an inch above the table, and the LEDs are the only connections between these two sets, and the only connections are the black circles.

You are correct, electricity will follow a path of least resistance to the greatest extent. In the schematic, there is no path of least resistance, i.e. short-circuit, between row and column wires, since no junctions are indicated. If you have interconnected the row and column wires, disconnect them now. Then, as you will see, the only path that allows electricity flow is through the respective diode / LED.