On the largest eigenvalues of trees

Deviation Inequalities on Largest Eigenvalues

In these notes, we survey developments on the asymptotic behavior of the largest eigenvalues of random matrix and random growth models, and describe the corresponding known non-asymptotic exponential bounds. We then discuss some elementary and accessible tools from measure concentration and functional analysis to reach some of these quantitative inequalities at the correct small deviation rate ...

On the sum of the two largest Laplacian eigenvalues of trees

On the largest k th eigenvalues of trees with n ≡ 0 ( mod k ) ø

We consider the only remaining unsolved case n ≡ 0 (mod k) for the largest kth eigenvalue of trees with n vertices. In 1995, Jia-yu Shao gave complete solutions for the cases k = 2, 3, 4, 5, 6 and provided some necessary conditions for extremal trees in general cases (cf. [Linear Algebra Appl. 221 (1995) 131]). We further improve Shao’s necessary condition in this paper, which can be seen as th...

Bounds on the Largest of Minimum Degree Eigenvalues of Graphs

Let G be a simple graph and let its vertex set be V (G) = {v1, v2, ..., vn}. The adjacency matrix A(G) of the graph G is a square matrix of order n whose (i, j)-entry is equal to unity if the vertices vi and vj are adjacent, and is equal to zero otherwise. The eigenvalues λ1, λ2, ..., λn of A(G), assumed in non increasing order, are the eigenvalues of the graph G. The energy of G was first defi...

On Multiple Eigenvalues of Trees

Let T be a tree of order n > 6 with μ as a positive eigenvalue of multiplicity k. Star complements are used to show that (i) if k > n/3 then μ = 1, (ii) if μ = 1 then, without restriction on k, T has k+ 1 pendant edges that form an induced matching. The results are used to identify the trees with a non-zero eigenvalue of maximum possible multiplicity.