A Positive Combinatorial Formula for the Complexity of the $q$-Analog of the $n$-Cube
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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 $q$-analog $C_q(n)$ 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 $c(C(n))$.
2015 ◽
Vol 92
(3)
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pp. 365-373
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2010 ◽
Vol 19
(06)
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pp. 765-782
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1981 ◽
Vol 31
(2)
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pp. 240-248
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2010 ◽
Vol 20
(1)
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pp. 11-25
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1998 ◽
Vol 179
(1-3)
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pp. 155-166
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2016 ◽
Vol 25
(09)
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pp. 1641005