Proofs of Li, Wang and Guo's conjectures on (signless) Laplacian spectral radius

Abstract

Let $μ(G)$ and $q(G)$ be the Laplacian and signless Laplacian spectral radii of a graph $G$. Li, Wang and Guo (2026) conjectured that, if $H$ with $|V(H)|\geq 1$ is an induced subgraph of a graph $G$, then $μ(G)\le\max\limits_{v\in V(H)}μ(H_v^t)$ and $q(G)\le\max\limits_{v\in V(H)}q(H_v^t)$, where $t=e(G)-e(H)$ and $H_v^t$ is the graph obtained from $H$ by adding $t$ pendant edges to $v\in V(H)$. We prove this conjecture by establishing a spectral comparison theorem for a broader family of positive semidefinite graph matrices which includes the Laplacian and signless Laplacian matrices. We also characterize all equality cases for the prescribed subgraph $H$ with $e(H)\geq 1$ and $t\geq 1$.