On the Laplacian Spectral Radius of Unicyclic Graphs with Fixed Diameter

Abstract

The set of unicyclic graphs with \(n\) vertices and diameter \(d\) is denoted by \(\mathcal{U}_{n,d}\). For \(3 \leq i \leq d\), let \(P_{n-d-1}(i)\) be the graph obtained from path \(P_{d+1}: v_1 v_2 \ldots v_{d+1}\) by adding \(n-d-1\) pendant edges at \(v_i\), and \(U_{n-d-2}(i)\) be the graph obtained from \(P_{n-d-1}(i)\) by joining \(v_{i-2}\) and a pendant neighbor of \(v_{i}\). In this paper, we determine all unicyclic graphs in \(\mathcal{U}_{n,d}\) whose largest Laplacian eigenvalue is greater than \(n-d+2\). For \(n-d \geq 6\) and \(G \in \mathcal{U}_{n,d}\), we prove further that the largest Laplacian eigenvalue \(\mu(G) \leq \max\{\lambda(U_{n,d-2}(i)) \mid 3 \leq i \leq d\}\), and conjecture that \(\mathcal{U}_{n,d}.\) is the unique graph which has the greatest value of the greatest Laplacian eigenvalue in \(\mathcal{U}_{n,d}\). We also prove that the conjecture is true for \(3 \leq d \leq 6\).