Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees

Abstract

Let G be a simple graph, its Laplacian matrix is the difference of the diagonal matrix of its degrees and its adjacency matrix. Denote its eigenvalues by μ ( G ) = μ 1 ( G ) ⩾ μ 2 ( G ) ⩾ ⋯ ⩾ μ n ( G ) = 0 . A vertex of degree one is called a pendant vertex. Let T n , k be a tree with n vertices, which is obtained by adding paths P 1 , P 2 , … , P k of almost equal the number of its vertices to the pendant vertices of the star K 1 , k . In this paper, the following results are given: (1) Let T be a tree with n vertices and k pendant vertices. Then μ ( T ) ⩽ μ ( T n , k ) , where equality holds if and only if T is isomorphic to T n , k . (2) Let G be a simple connected bipartite graph with degrees d 1 , d 2 , … , d n . Then μ ( G ) ⩾ 2 1 n ∑ i = 1 n d i 2 , where equality holds if and only if G is a regular connected bipartite graph. (3) Let G be a simple connected bipartite graph with vertices v 1 , v 2 , … , v n and their degrees d 1 , d 2 , … , d n . Then μ ( G ) ⩾ 2 + 1 m ∑ v i ∼ v j , i < j ( d i + d j - 2 ) 2 , where m is the edge number of G and equality holds if and only if G is either a regular connected bipartite graph or a semiregular connected bipartite graph or the path with four vertices.