Signless Laplacian spectral characterization of line graphs of T-shape trees

A T-shape tree is a tree with exactly one vertex of maximum degree 3. The line graphs of the T-shape trees are triangles with a hanging path at each vertex. Let Ca,b,c be such a graph, where a, b and c are the lengths of the paths. In this paper, we show that line graphs of T-shape trees, with the sole exception of Ca,a,2a+1, are determined by the spectra of their signless Laplacian matrices. For the graph Ca,a,2a+1 we identify the unique non-isomorphic graph sharing the same signless Laplacian characteristic polynomial.