Graph Theory · Electric Circuits · GATE EE

The transfer function of a real system, H(s), is given as:

$$H(s) = {{As + B} \over {{s^2} + Cs + D}}$$,

where A, B, C and D are positive constants. This system cannot operate as

The number of chords in the graph of the given circuit will be

The graph of an electrical network has $$N$$ nodes and $$B$$ branches. The number of links, $$L$$, with respect to the choice of a tree, is given by

Figure shows $$a$$ $$b$$ $$c$$. resistive network and its graph is drawn aside. $$A$$ 'proper tree' chosen for analyzing the network will contain the edges :

The matrix $$A$$ given below is the node incidence matrix of a network. The columns correspond to braches of the network while the rows correspond to nodes. Let
$$V = {\left[ {{v_1}\,\,{v_2}....{v_6}} \right]^T}$$ denote the vector of branches voltages while
$${\rm I} = {\left[ {{i_1}\,{i_2}....{i_6}} \right]^T}$$ that of branch currents. The vector $$E = {\left[ {{e_1}\,{e_2}\,\,{e_3}\,{e_4}} \right]^T}$$ denotes the vector of node voltages relative to a common ground. $$$A = \left[ {\matrix{ 1 & 1 & 1 & 0 & 0 & 0 \cr 0 & { - 1} & 0 & { - 1} & 1 & 0 \cr { - 1} & 0 & 0 & 0 & { - 1} & { - 1} \cr 0 & 0 & { - 1} & 1 & 0 & 1 \cr } } \right]$$$

Which of the following statements is true?