There are 11 integral trees with largest eigenvalue 3.

We classify the growth of a k-regular sequence based on information from its k-kernel. In order to provide such a classification, we introduce the notion of a growth exponent for k-regular sequences and show that this exponent is equal to the joint spectral radius of any set of a special class of matrices determined by the k-kernel.

Recently some important results have been proved showing that the gap between the largest eigenvalue A: of a finite regular graph of valency k and its second eigenvalue is related to expansion properties of the graph [1]. In this paper we investigate infinite graphs and show that in this case the expansion properties are related to the spectral radius of the graph. First we introduce necessary ...

In this paper, the trees with the largest Dirichlet spectral radius among all trees with a given degree sequence are characterized. Moreover, the extremal graphs having the largest Dirichlet spectral radius are obtained in the set of all trees of order n with a given number of pendant vertices.

Using a result linking convexity and irreducibility of matrix sets it is shown that the generalized spectral radius of a compact set of matrices is a strictly increasing function of the set in a very natural sense. As an application some consequences of this property in the area of time-varying stability radii are discussed. In particular, using the implicit function theorem sufficient conditio...

A b s t r a c t. Let K3 and K ′ 3 be two complete graphs of order 3 with disjoint vertex sets. Let B∗ n(0) be the 5-vertex graph, obtained by identifying a vertex of K3 with a vertex of K ′ 3 . Let B∗∗ n (0) be the 4-vertex graph, obtained by identifying two vertices of K3 each with a vertex of K ′ 3 . Let B∗ n(k) be graph of order n , obtained by attaching k paths of almost equal length to the...

A decomposition result for planar graphs is used to prove that the spectral radius of a planar graph on n vertices is less than 4 + 3(n 3) Moreover, the spectral radius of an outerplanar graph on n vertices is less than 1 + JZ+&-X

Let G be a graph with n vertices, m edges, girth g, and spectral radius μ. Then

If ρ(A) > 1, then lim n→∞ ‖A‖ =∞. Proof. Recall that A = CJC−1 for a matrix J in Jordan normal form and regular C, and that A = CJnC−1. If ρ(A) = ρ(J) < 1, then J converges to the 0 matrix, and thus A converges to the zero matrix as well. If ρ(A) > 1, then J has a diagonal entry (J)ii = λ n for an eigenvalue λ such that |λ| > 1, and if v is the i-th column of C and v′ the i-th row of C−1, then ...