Let $G$ be a graph with $n$ vertices. Denote by $L(G)$ the Laplacian matrix of $G$ and $0=\lambda_1\leqslant\lambda_2\leqslant...\leqslant\lambda_n$ its spectrum. The number $\lambda_2$ is called the algebraic connectivity of $G$ and is denoted by $a(G)$.

In my research I need the value of $a(P_n)$, where $P_n$ is the path with $n$ vertices. I know (and I have a proof) that $a(P_n)=4\sin^2(\pi/(2n))$. It seems that this result is very well-known and classical, but I cannot find a proof in the literature (in some papers the authors just write $a(P_n)=4\sin^2(\pi/(2n))$ without proof).

Thus I would be very grateful for the reference to a proof of the result (if the proof exists in the literature).

  • $\begingroup$ here sciencedirect.com/science/article/pii/… it is claimed that the papers [5,6] contain what you need $\endgroup$ Jul 19, 2019 at 19:46
  • $\begingroup$ [5] R. Grone, R. Merris, V.S.Sunder, The Laplacian spectrum of a graph. $\endgroup$ Jul 19, 2019 at 20:35
  • $\begingroup$ They refer to the paper of Fiedler: M. Fiedler, Algebraic connectivity of graphs, 1973. But there is NO proof in the paper of Fiedler. $\endgroup$ Jul 19, 2019 at 20:45
  • $\begingroup$ [6] R. Grone, R.Merris, The Laplacian spectrum of a graph II, 1994. I did not find the result on the algebraic connectivity of the path in the paper. $\endgroup$ Jul 19, 2019 at 21:02
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    $\begingroup$ Do you need the reference to any proof or to the first published proof ever? You may take the proof say here ocw.mit.edu/courses/mathematics/… if you need a book I would check that of C. Godsil, G. F. Royle (Algebraic graph theory) or Fan R. K. Chung (Spectral graph theory) $\endgroup$ Jul 19, 2019 at 22:41


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