On the \(Q\)-index and Index of Triangle-Free Quasi-Tree Graphs

Abstract

A connected graph \(G\) is called a quasi-tree graph, if there exists \(v_0 \in V(G)\) such that \(G – v_0\) is a tree. In this paper, among all triangle-free quasi-tree graphs of order \(n\) with \(G – v_0\) being a tree and \(d(v_0) = d(v_0)\), we determine the maximal and the second maximal signless Laplacian spectral radii together with the corresponding extremal graphs. By an analogous manner, we obtain similar results on the spectral radius of triangle-free quasi-tree graphs.