The transfer function of the given signal flow graph is

Concept:

A signal flow graph represents the relationships between the variables of a set of linear algebraic equations.

Manson's gain formula:

\(T = \mathop \sum \limits_{k = 1}^k \frac{{{P_k}{{\rm{\Delta }}_k}}}{{\rm{\Delta }}}\)

• Where Pk is the forward path transmittance of kth in the path from a specified input to an output node. In arresting Pk no node should be encountered more than once.
• Δ is called the signal flow graph determinant.
• Δ = 1 – (sum of all individual loop transmittances) + (sum of loop transmittance products of all possible pairs of non-touching loops) – (sum of loop transmittance products of all possible triplets of non-touching loops) + (……) – (……)
• Δ k is the factor associated with the concerned path and involves all closed loop in the graph which are isolated from the forward path under consideration.
• The path factor Δk for the kth path is equal to the value of the grab determinant of its signal flow graph, which exists after erasing the Kth path from the graph.
  
  Given:​
  
  
  The transfer function \(\frac{Y(s)}{X(s)} = \frac{1}{1-G}\)