Algebraic connectivity of lollipop graphs: a new approach (Q2875694)

This paper gives a new, alternative proof for a result of \textit{J.-M. Guo} et al. [Linear Algebra Appl. 434, No. 10, 2204--2210 (2011; Zbl 1227.05182)] on the algebraic connectivity (the value of the second-smallest eigenvalue of the Laplacian matrix) of lollipop graphs: a lollipop graph \(C_{n,g}\) is obtained by attaching a cycle of length \(g\) to one end of a path on \(n\mathrm{-}g\) vertices.NEWLINENEWLINESpecifically, it is shown that (where \(a\) denotes the algebraic connectivity) NEWLINE\[NEWLINEa(C_{n,3}) < a(C_{n,4}) < \cdots < a(C_{n,n-1}) < a(C_n) = 2\left(1- \cos \frac{2\pi}{n}\right)NEWLINE\]NEWLINE for all \(n \geq 3\).