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?

Question

If the reduced incidence matrix network is given as

[latex] 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} ight] [/latex]

then what are possible number of trees?

Accepted Answer

[latex] \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} ight]\left[\begin{array}{rrr} -1 & 0 & 0 \ 0 & 1 & 0 \ 0 & -1 & 0 \ 1 & -1 & 0 \ 1 & 0 & -1 \ 0 & 0 & 1 \end{array} ight] \ &=\left[\begin{array}{rrr} 3 & -1 & -1 \ -1 & 3 & -1 \ -1 & -1 & 3 \end{array} ight] \end{aligned} [/latex]

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

Chapter 1

Q. 1.32

Q. 1.32

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