A Note on the Signless Laplacian Spectral Ordering of Graphs with Given Size

Abstract:

      For a simple undirected graph $G$ with fixed size $m\ge 2k~(k \in \mathbb{Z}^{+})$ and maximum degree $\Delta(G)\le m-k$, we give an upper bound on the signless Laplacian spectral radius $q(G)$ of $G$. For two connected graphs $G_1$ and $G_2$ with size $m\ge 8$, employing this upper bound, we prove that $q(G_1)>q(G_2)$ if $\Delta(G_1)>\Delta(G_2)+1$ and $\Delta(G_1)\ge \frac{m}{2}+2$. For triangle-free graphs, we prove two stronger results. As an application, we completely characterize the graph with maximal signless Laplacian spectral radius among all graphs with size $m$ and circumference $c$ for $m\ge \max\{ 2c, c+9\}$, which partially answers the question proposed by Chen et al. in [Linear Algebra Appl., 2022, 645: 123--136].