Question 1.P.32: If the reduced incidence matrix network is given as A = [ -1......

If the reduced incidence matrix network is given as

A=\left[\begin{array}{rrrrrr} -1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & -1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 & -1 & 1 \end{array}\right]

then what are possible number of trees?

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\begin{aligned} & A^{ T }=\left[\begin{array}{rrrrrr} -1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & -1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 & -1 & 1 \end{array}\right]\left[\begin{array}{rrr} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 0 \\ 1 & -1 & 0 \\ 1 & 0 & -1 \\ 0 & 0 & 1 \end{array}\right] \\ &=\left[\begin{array}{rrr} 3 & -1 & -1 \\ -1 & 3 & -1 \\ -1 & -1 & 3 \end{array}\right] \end{aligned}

The number of trees is det \left[A_{ r } A_{ r }^T\right]=3(9-1)+ 1(−3 − 1) − 1 = 24 − 4 − 4 = 16

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