Signless Laplacian spectral conditions for extremal quadrilateral 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 such graphs. The main theorem states that for integers $k\ge 0$ and graphs $G$ with $m$ edges where $m\ge max\{7k+31,k^2+8(k+1)\}$, if $q\left(G\right)\ge q\left(S_{m,k+1}^{+}\right)$, 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 related previous work on star embeddings by Wang and Guo (2024), and completes the $Q$-spectral counterpart to Wang’s adjacency spectral theorem for 4-cycle containment (Wang, 2022). 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.