Arduino on Breadboard troubleshoot, the importance of Capacitors

I have an Arduino on a breadboard and having an issue with the voltage on the digital pin 13 outputing an low voltage when connect to a volt regulator.  Why is this happening? because when the breadboard is connected to an Arduino UNO power that is connected to the same power supply it works perfectly fine.  See images and descriptions below.  Thank you for the help.

Notes:
Power wires are connected at the bottom of the breadboard
Components:

Arduino on breadboard
Power Jack
9v – 300mA Power
L7805CV 5voltage regulator

Arduino Code:
Set to output LIGHT ON on digital pin 13

DSC02219
The voltage on the rails when breadboard is connected to power is 5 volts

DSC02221
The voltage on digital pin 13 when connected to power is 0.370 volts (why?)

 

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The voltage on the rails when connected to arduino power is 5 volts

 

DSC02223
The voltage on digital pin 13 when connected to arduino power is 4.74 volts

Why is the voltage on digital pin 13 not the same when connected to an Arduino Uno Power to an Power supply using a voltage regulator?

 

Answer:
Its very important to add a electrolytic capacitor to the input and output caps of the voltage regulator because this will help SMOOTH out the ripple the battery pack/regulator is putting out.

I only added a 22uF electrolytic capacitor to the rails and it worked.  See picture below.

DSC02224 copy

Reference