# How to prove this fundamental relationship $b=\ell+n-1$?

How to prove this fundamental relationship?

In a network or circuit, number of loop, nodes and branches has to satisfy the following fundamental relationship: $$b=\ell+n-1,$$ where, $b$ = number of branches, $\ell$ = number of loops, and $n$ = number of nodes. 

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## migrated from physics.stackexchange.comJun 10 '13 at 1:34

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@ Qmechanic Thank you for the reply. –  user7777777 Jun 9 '13 at 17:22

## 1 Answer

This is true for a connected planar circuit (usually called a graph). To prove it just think about what happens to the value of $l + n - b$ whenever you reduce the circuit by removing a branch or a node, without dividing it into two parts. If you look at all possible cases you will find that this number always stays the same. Eventually you will be left with just one node so the value must be $1$

(Please note that since this has been moved from physics to maths where I am not registered I cannot reply to any further comments)

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@ Philip Gibbs Thank you very much. "Eventually you will be left with just one node..." Is it OK to view this like- Eventually you will be left with just one loop...? –  user7777777 Jun 9 '13 at 17:55
The loops are areas surrounded by branches (better known as edges), so there would be no loops at the end. –  Philip Gibbs Jun 9 '13 at 20:30
@ Philip Gibbs However node is defined as a connection point between two or more branches. So how there could be nodes at the end? Thanks in advance –  proofy Jun 13 '13 at 17:50