Algebraic connectivity of the path $P_n$

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).

• here sciencedirect.com/science/article/pii/… it is claimed that the papers [5,6] contain what you need – Fedor Petrov Jul 19 '19 at 19:46
• [5] R. Grone, R. Merris, V.S.Sunder, The Laplacian spectrum of a graph. – Ivan Feshchenko Jul 19 '19 at 20:35
• They refer to the paper of Fiedler: M. Fiedler, Algebraic connectivity of graphs, 1973. But there is NO proof in the paper of Fiedler. – Ivan Feshchenko Jul 19 '19 at 20:45
• [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. – Ivan Feshchenko Jul 19 '19 at 21:02
• 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) – Fedor Petrov Jul 19 '19 at 22:41