A positive combinatorial formula for the complexity of the \(q\)-analog of the \(n\)-cube (Q426914)

Summary: The number of spanning trees of a graph \(G\) is called the complexity of \(G\). A classical result in algebraic graph theory explicitly diagonalizes the Laplacian of the \(n\)-cube \(C(n)\) and yields, using the Matrix-Tree theorem, an explicit formula for \(c(C(n))\). In this paper we explicitly block diagonalize the Laplacian of the \(q\)-ana\(\log C_q(n)\) of \(C(n)\) and use this, along with the Matrix-Tree theorem, to give a positive combinatorial formula for \(c(C_q(n))\). We also explain how setting \(q=1\) in the formula for \(c(C_q(n))\) recovers the formula for \(c(C(n))\).