Pattern Recognition on Oriented Matroids

13.3Symmetric Cycles in the Hypercube Graphs

In this section we turn to an investigation of the combinatorial properties of the vertex set {1, –1}t of the hypercube graph H(t, 2) and of its symmetric cycles.

Let R be a symmetric cycle in H(t, 2), with its distance vectors zT,R associated with the topes T ∈ {1, –1}t.

According to eq. (13.5), we have

$\begin{array}{l}{\displaystyle \sum _{T\in {\left\{1,-1\right\}}^t}\left|Q\left(T,R\right)\right|}\\ ={\displaystyle \sum _{T\in {\left\{1,-1\right\}}^t}\left(t-\frac{1}{4}{\displaystyle \sum _{n=0}^{2t-1}{z}_{T,R}\left(n\right)\left(z_{T,R}\left(n\right)-z_{T,R}\left(\left(n+2\right)\mathrm{mod}2t\right)\right)}\right)}\\ =2^{t}t-\frac{1}{4}{\displaystyle \sum _{n=0}^{2t-1}{\displaystyle \sum _{T\in {\left\{1,-1\right\}}^t}{z}_{T,R}\left(n\right)\left(z_{T,R}\left(n\right)-z_{T,R}\left(\left(n+2\right)\mathrm{mod}2t\right)\right)}}\\ =2^{t}t-\frac{1}{2}t{\displaystyle \sum _{T\in {\left\{1,-1\right\}}^t}{z}_{T,R}(}\end{array}$