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Energy of a hypercube and its complement. (English) Zbl 1251.05096

Summary: Let \(\bar Q_n\) denote the complement of \(n\)-dimensional hypercube \(Q_n\), and let \(E(G)\) and \(LE(G)\) denote, respectively, the (ordinary) energy and Laplacian energy of a graph \(G\).
We obtain
\[ \begin{aligned} LE(Q_n) &= E(Q_n) = 2\lceil\frac{n}{2}\rceil\binom n{\lceil\frac{n}{2}\rceil}\\ \text{and} LE(\bar Q_n) &= E(\bar Q_n) = (n + 1) \binom n{\lceil\frac{n}{2}\rceil} + 2^n - 2^n - 2, \end{aligned} \]
where \(n\geq 1\).

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C65 Hypergraphs
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