Fundamental loop: A fundamental loop is a closed path of a given graph with only one Link and the rest of them as twigs. The number of fundamental loops for any given graph = b – (n – 1) = number of Links These fundamental loop currents are called Tie set currents and the orientation of the tie set currents governed by the link. = = +1, If jth branch current is incident at ith tie set current at oriented in the same direction. = –1, if jth branch current is incident at ith tie set current at oriented in the opposite direction. = 0, If jth branch current is not incident with ith tie set current. For the above graph, the tie set matrix is given by [ alt="F3 U.B Madhu 24.04.20 D 25" src="//storage.googleapis.com/tb-img/production/20/04/F3_U.B_Madhu_24.04.20_D%2025.png" style="width: 276px; height: 104px;"> We can rearrange the matrix as given below. [ alt="F3 U.B Madhu 24.04.20 D 26" src="//storage.googleapis.com/tb-img/production/20/04/F3_U.B_Madhu_24.04.20_D%2026.png" style="width: 275px; height: 117px;"> Tie set currents and branch currents together form an identity matrix as marked in the above Tie set matrix. Now, from the above matrix, it is clear that, The submatrix corresponding to twigs (Bt)­ is not an identity matrix The submatrix corresponding to twigs (Bl)­ is an identity matrix and a rectangular matrix The rank of tie set matrix is b – (n – 1) i.e. 3 for the above matrix