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

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 $1$
@ 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