The Laplacian Spectral Radius of Unicyclic Graphs with \(k\) Pendent Vertices

Abstract

Let \(\mathcal{U}_n(k)\) denote the set of all unicyclic graphs on \(n\) vertices with \(k\) (\(k \geq 1\)) pendant vertices. Let \(\diamondsuit_4^k\) be the graph on \(n\) vertices obtained from \(C_4\) by attaching \(k\) paths of almost equal lengths at the same vertex. In this paper, we prove that \(\diamondsuit_4^k\) is the unique graph with the largest Laplacian spectral radius among all the graphs in \(\mathcal{U}_n(k)\), when \(n \geq k + 4\).