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

AbstractLet 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)=μ1(G)⩾μ2(G)⩾⋯⩾μn(G)=0. A vertex of degree one is called a pendant vertex. Let Tn,k be a tree with n vertices, which is obtained by adding paths P1,P2,…,Pk of almost equal the number of its vertices to the pendant vertices of the star K1,k. In this paper, the following results are given:(1) Let T be a tree with n vertices and k pendant vertices. Thenμ(T)⩽μ(Tn,k),where equality holds if and only if T is isomorphic to Tn,k.(2) Let G be a simple connected bipartite graph with degrees d1,d2,…,dn. Thenμ(G)⩾21n∑i=1ndi2,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 v1,v2,…,vn and their degrees d1,d2,…,dn. Thenμ(G)⩾2+1m∑vi∼vj,i<j(di+dj-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

Largest Laplacian Eigenvalue and Degree Sequences of Trees

We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum eigenvalue in such classes of trees is strictly monotone with respect to majorization.Comment: 9 pages, 2 figure

Walks and the spectral radius of graphs

We give upper and lower bounds on the spectral radius of a graph in terms of the number of walks. We generalize a number of known results.Comment: Corrections were made in Theorems 5 and 11 (the new numbers are different), following a remark of professor Yaoping Ho

Eigenvalue bounds for the signless laplacian

We extend our previous survey of properties of spectra of signless Laplacians of graphs. Some new bounds for eigenvalues are given, and the main result concerns the graphs whose largest eigenvalue is maximal among the graphs with fixed numbers of vertices and edges. The results are presented in the context of a number of computer-generated conjectures

On the Signless Laplacian Spectral Radius of Bicyclic Graphs with Perfect Matchings

The graph with the largest signless Laplacian spectral radius among all bicyclic graphs with perfect matchings is determined

Properties and Applications of the Eigenvector Corresponding to the Laplacian Spectral Radius of a Graph

We study the properties of the eigenvector corresponding to the Laplacian spectral radius of a graph and show some applications. We obtain some results on the Laplacian spectral radius of a graph by grafting and adding edges. We also determine the structure of the maximal Laplacian spectrum tree among trees with n vertices and k pendant vertices (n, k fixed), and the upper bound of the Laplacian spectral radius of some trees