on connected bipartite $q$-integral graphs

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.