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 C a,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 C a,a,2a+1 , are determined by the spectra of their signless Laplacian matrices. For the graph C a,a,2a+1 we identify the unique non-isomorphic graph sharing the same signless Laplacian characteristic polynomial.
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