How it works
For a graph with N nodes and B branches: number of tree branches = N−1 (twigs), number of links (chords) = B−N+1. A tree connects all nodes with no closed loop; a cotree consists of the remaining links. The reduced incidence matrix A is (N−1)×B, where aij = +1 if branch j leaves node i, −1 if it enters, 0 otherwise. Fundamental cut-set matrix Q is (N−1)×B: each row corresponds to one twig and the fundamental cut-set formed when that twig is removed. Fundamental tie-set (loop) matrix B is (B−N+1)×B: each row is one link and its fundamental loop with tree branches. KCL in matrix form: A·I_b = 0; KVL: B·V_b = 0.
Key points to remember
Number of independent loop equations = B − N + 1 (number of links), and number of independent node equations = N − 1 — for a circuit with 5 nodes and 8 branches, this gives 4 node and 4 loop equations. The rank of the incidence matrix equals N−1 for a connected graph. Duality in network graphs: the number of tree branches and links swap when the dual network is formed, and incidence matrix columns correspond to cut-set matrix rows in the dual. The term "planar graph" means the graph can be drawn without edge crossings — only planar networks can be analysed by mesh analysis without additional loop selection care. Network topology does not depend on element values, only on the connection pattern.
Exam tip
The examiner always asks you to draw the graph of a given circuit with 5 or 6 elements, identify a tree, write the reduced incidence matrix, and state the number of independent KVL and KCL equations — count nodes and branches first, apply the formulas, then draw the tree carefully.