# On the signless Laplacian spectral characterization of the line graphs of *T*-shape trees

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## Abstract

A graph is determined by its signless Laplacian spectrum if no other nonisomorphic 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 *L*(*T*(*a*, *b*, *c*)) except *L*(*T*(*t*, *t*, 2*t*+1)) (*t* ⩾ 1) are *DQS*, and determine the graphs which have the same signless Laplacian spectrum as *L*(*T*(*t*, *t*, 2*t* + 1)). Let µ_{1}(*G*) be the maximum signless Laplacian eigenvalue of the graph *G*. We give the limit of µ_{1}(*L*(*T*(*a*, *b*, *c*))), too.

## Keywords

signless Laplacian spectrum cospectral graphs*T*-shape tree

## MSC 2010

05C50 15A18## Preview

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© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2014