Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 11 3 2026 09 01 On Connected Graphs with Integer-Valued Q-Spectral Radius 767 786 14847 10.22049/cco.2024.29278.1922 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 Smrati Pandey Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi-221005, India Journal Article 2023 12 17 The $Q$-eigenvalues are the eigenvalues of the signless Laplacian matrix $Q(G)$ of a graph $G$, and the largest $Q$-eigenvalue is known as the $Q$-spectral radius $q(G)$ of $G$. The edge-degree of an edge is defined as the number of edges adjacent to it. In this article, we characterize the structure of simple connected graphs having integral $Q$-spectral radius. We show that the necessary and sufficient condition for such graphs to contain either a double star $\mathcal{S}_{r}^{2}$ or its variation $\mathcal{S}_{r}^{2,1}$ (having exactly one common neighbor between the central vertices) as a subgraph is that the maximum edge-degree is $2r$, where $r= q(G) -3$. In particular, we characterize all graphs that contain only double star as a subgraph when $q(G)$ equals $8$ and $9$. Further, we characterize all the connected edge-non-regular graphs with a maximum edge-degree equal to $4$ whose minimum  $Q$-eigenvalue does not belong to the open interval $(0,1)$ and has an integral $Q$-spectral radius. https://comb-opt.azaruniv.ac.ir/article_14847_92c0c4c101eb3908bc4f77ca725a7225.pdf