If b and n be the number of branches and number of nodes, respectively, and s be the number of separate parts, then the number of independent loop equations will be:

Concept:

A loop is said to be independent if it contains at least one branch which is not a part of any other independent loop. Independent loops or paths result in independent sets of equations.

Branch: An element or edge of a tree of a connected graph is called a branch.

Node: Nodes are the vertices in the graph.

Separate part: A connected subgraph of a graph is said to be a separate part if it contains no vertices in common with its complement i.e the remaining subgraph. 

Solution:

If b, n, and s are the number of branches, nodes, and separate parts of the network respectively, then the number of linearly independent mesh equations,

m = b - n + s

Additional Information The number of independent node equations for the graph is given by (n-1) where n is the number of nodes.