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

Guoping WangGuangquan GuoLi Min — 2014

Czechoslovak Mathematical Journal

A graph is determined by its signless Laplacian spectrum if no other non-isomorphic graph has the same signless Laplacian spectrum (simply G is D Q S ). 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 ( T ( a , b , c ) ) except ( T ( t , t , 2 t + 1 ) ) ( t 1 ) are D Q S , and determine the graphs which have the same signless Laplacian spectrum as ( 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 ( ( T ( a , b , c ) ) ) , too.

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