Connected graphs having maximum signless Laplacian spectral radius

Abstract

Let $\rho (G)$ denote the signless Laplacian spectral radius of a graph $G$, and let ${\mathcal{C}}_{n,m}$ be the family of connected graphs of order $n$ and size $m$. A graph is $Q$-maximal if it maximizes $\rho (G)$ over this family. Motivated by a question of Chang and Tam (2011), we study this extremal problem using scalar identities and coronal methods for threshold graphs. From the Perron equations, we derive the identity $S_G(\lambda )=T_{a,b}(\lambda )$ at $\lambda =\rho (G)$, where $a$ and $b$ are the total numbers of zero-vertices and one-vertices. This yields a one-point comparison principle: for a connected threshold graph $H$ with the same block totals, $S_H(z)>T_{a,b}(z)$ at some $z\ge n$ implies $\rho (H)>z$, allowing spectral comparison without estimating individual Perron coordinates. We prove that every $Q$-maximal threshold graph has character at most three, so its vertex-sequence contains at most three zero--one block pairs. We also show that the quasi-star graph is $Q$-maximal whenever $n\ge 5$ and $n-1\le m<(n^2+10n-8)/8$. For certain dense parameter pairs, we construct connected threshold graphs with signless Laplacian spectral radii strictly larger than those of both the quasi-star and the connected quasi-complete graph. Thus the corresponding two-candidate assertion for connected graphs fails in general.