Signless Laplacian spectral conditions: Forbidden $4$-cycle and star embeddings

The signless Laplacian spectral radius has emerged as a crucial spectral parameter in network science. This paper establishes new extremal results in spectral graph theory by investigating the signless Laplacian spectral radius ($Q$-index) of graphs with forbidden subgraphs. We present a $Q$-spectral analog of classical Nosal-type theorems, providing sharp conditions that guarantee the existence of either a $4$-cycle or a large star $K_{1,m-k}$ in a graph. The main theorem states that for integers $k \geq 0$ and graphs $G$ with size $m \geq \max\{7k+31, k^2+8(k+1)\}$, if $q(G) \geq q(S^+_{m,k+1})$, then $G$ must contain a $4$-cycle or $K_{1,m-k}$, unless $G$ is isomorphic to the extremal graph $S^+_{m,k+1}$ formed by adding $k+1$ independent edges to the star $K_{1,m-k-1}$. This result refines previous work on star embeddings by Wang and Guo [Journal of Algebraic Combinatorics, 59 (2024) 213--224], and completes the $Q$-spectral counterpart to Wang's adjacency spectral theorem for $4$-cycle containment [Discrete Math., 345 (2022) 112973]. Our analysis reveals new insights into how signless Laplacian eigenvalues encode graph structure, with tight bounds demonstrated through explicit extremal graph constructions and asymptotic analysis.