Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 10 4 2025 12 01 On connected bipartite $Q$-integral graphs 729 742 14698 10.22049/cco.2024.29215.1895 EN Jesmina Pervin Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi-221005, India Lavanya Selvaganesh Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi-221005, India Journal Article 2023 11 22 A graph $G$ is said to be $H$-free if $G$ does not contain $H$ as an induced subgraph. Let $\mathcal{S}_{n}^2(m)$ be a \textit{variation of double star $\mathcal{S}_{n}^2$} obtained by adding m (<=n) disjoint edges between the pendant vertices which are at distance 3 in $\mathcal{S}_{n}^2$. A graph having integer eigenvalues for its signless Laplacian matrix is known as a Q-integral graph. The Q-spectral radius of a graph is the largest eigenvalue of its signless Laplacian. Any connected Q-integral graph G with Q-spectral radius 7 and maximum edge-degree 8 is either $K_{1,4}\square K_2$ or contains $\mathcal{S}_{4}^2(0)$ as an induced subgraph or is a bipartite graph having at least one of the induced subgraphs $\mathcal{S}_{4}^2(m)$, (m=1, 2, 3). In this article, we improve this result by showing that every connected Q-integral graph G having Q-spectral radius 7, maximum edge-degree 8 is always bipartite and $\mathcal{S}_{4}^2(3)$-free. https://comb-opt.azaruniv.ac.ir/article_14698_0ae966057791b15fb5438f2f90d99bf6.pdf