This article is concerned with the design of boundary control laws for stabilizing systems of 2 × 2 first order quasi-linear hyperbolic PDEs. A new graph representation of such systems represents the interactions between the characteristic curves and the boundary control laws, the invariant graph, is introduced. The structure of the invariant graph is used to design stabilizing control laws and...

For a simple connected graph G with n-vertices having Laplacian eigenvalues μ1, μ2, . . . , μn−1, μn = 0, and signless Laplacian eigenvalues q1, q2, . . . , qn, the Laplacian-energy-like invariant(LEL) and the incidence energy (IE) of a graph G are respectively defined as LEL(G) = ∑n−1 i=1 √ μi and IE(G) = ∑n i=1 √ qi. In this paper, we obtain some sharp lower and upper bounds for the Laplacian...

In this paper we consider a structured linear system represented by means of a directed graph. We present a graph theoretic method to compute the generic number of invariant zeros of the corresponding Rosenbrock matrix. The method is based on a fundamental decomposition of the directed graph representing the structured system. The generic number of invariant zeros is important in generic versio...

The diameter of a graph G is the maximal distance between two vertices of G. A graph G is said to be diameter-edge-invariant, if d(G−e) = d(G) for all its edges, diametervertex-invariant, if d(G − v) = d(G) for all its vertices and diameter-adding-invariant if d(G+ e) = d(e) for all edges of the complement of the edge set of G. This paper describes some properties of such graphs and gives sever...

A spanning subgraph of a graph G is called a [0,2]-factor of G, if 0 ( ) 2 d x   for ( ) x V G   . is a union of some disjoint cycles , paths and isolate vertices , that span the graph G. It is easy to get a [0,2]-factor of G and there would be many of [0,2]-factors for a G. A characteristic number for a [0,2]-factor, which reflect the number of the paths and isolate vertices in it,. The [0...

We classify the gauge-invariant ideals in the C∗-algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gauge-invariant primitive ideals in terms of the structural properties of the graph, and describe the K-theory of the C∗-algebras of arbitrary infinite graphs.

We define a new invariant for shift equivalence of sofic shifts. This invariant, that we call the syntactic graph of a sofic shift, is the directed acyclic graph of characteristic groups of the non null regular D-classes of the syntactic semigroup of the shift.

A relationship between a new and an old graph invariant is established. The first invariant is connected to the 'sandglass conjecture' of [1]. The second one is graph entropy, an information theoretic functional, which is already known to be relevant in several combinatorial contexts.

Let H be a Hilbert space of analytic functions with multiplier algebra M(H), and let M = {(f, T1f...., Tn−1f) : f ∈ D} be an invariant graph subspace for M(H). Here n ≥ 2, D ⊆ H is a vector-subspace, Ti : D → H are linear transformations that commute with each multiplication operator Mφ ∈ M(H), and M is closed in H. In this paper we investigate the existence of nontrivial common invariant subsp...

Let G be an undirected simple graph with n vertices and m edges. Denote with |λ1| |λ2| · · · |λn| and |ρ1| |ρ2| · · · |ρn| absolute eigenvalues and Randić eigenvalues of G arranged in non-increasing order, respectively. Upper bound of graph invariant E(G) = ∑i=1 |λi| , and lower and upper bounds of invariant RE(G) =∑i=1 |ρi| are obtained in this paper.