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Let G = ( V , E ) be a simple connected graph. A pendant dominating set S of a graph G is a bi-pendant dominating set if V - S also contains pendant vertex. The least cardinality of the bi-pendant dominating set in G is called the bi-pendant domination number of G denoted by γ b p e ( G ). If G 1 , G 2 , G 3 , …, G n are connected edge disjoint sub graphs of G with E ( G ) = E ( G 1 ) ∪ E ( G 2 ) ∪ E ( G 3 ) …∪ E ( G n ), then G 1 , G 2 , G 3 , …, G n is said to be decomposition of G. In this paper, we define Ascending Bi-Pendant Domination Decomposition (ABPDD) and discuss the values of m in P m and C m which admits APBDD into n -parts.

LetmT [0, 2) be the number of Laplacian eigenvalues of a treeTin [0, 2), multiplicities included. We give best possible upper bounds formT [0, 2) using the parameters such as the number of pendant vertices, diameter, matching number, and domination number, and characterize the treesTof ordernwithmT [0, 2) =n− 1,n− 2, and ⌈ n 2 ⌉ , respectively, and in particular, show that m T [ 0 , 2 ) = ⌈ n 2 ⌉ if and only if the matching number ofTis ⌊ n 2 ⌋ . 2010Mathematics Subject Classification: 05C50, 05C35. Key words and phrases: Laplacian eigenvalues, Trees, Pendant vertex, Diameter, Matching number, Domination number.