Ordering Graphs with Given Size by Their Signless Laplacian Spectral Radii

Let q(G) denote the signless Laplacian spectral radius of a graph G. In this paper, we first give an upper bound on q(G) of a connected graph G with fixed size \(m\ge 3k(k \in {\mathbb {Z}}^{+})\) and maximum degree \(\Delta \le m-k\). For two connected graphs \(G_1\) and \(G_2\) with size \(m\ge 4\), employing this upper bound, we prove that \(q(G_1)>q(G_2)\) if \(\Delta (G_1)>\Delta (G_2)\) and \(\Delta (G_1)\ge \frac{2m}{3}+1\). As an application, we determine the first \(\lfloor d/2\rfloor \) graphs with the largest signless Laplacian spectral radius among all graphs with fixed size and diameter.