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Keywords:
signless Laplacian spectrum; cospectral graphs; $T$-shape tree
Summary:
A graph is determined by its signless Laplacian spectrum if no other non-isomorphic graph has the same signless Laplacian spectrum (simply $G$ is $DQS$). Let $T(a,b,c)$ denote the $T$-shape tree obtained by identifying the end vertices of three paths $P_{a+2}$, $P_{b+2}$ and $P_{c+2}$. We prove that its all line graphs $\mathcal {L}(T(a,b,c))$ except $\mathcal {L}(T(t,t,2t+1))$ ($t\geq 1$) are $DQS$, and determine the graphs which have the same signless Laplacian spectrum as $\mathcal {L}(T(t,t,2t+1))$. Let $\mu _1(G)$ be the maximum signless Laplacian eigenvalue of the graph $G$. We give the limit of $\mu _1(\mathcal {L}(T(a,b,c)))$, too.
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