On spectra of expansion graphs and matrix polynomials

Förster, K.-H.; Nagy, B.

doi:10.1016/s0024-3795(01)00595-x

Cited by 4 publications

(18 citation statements)

References 13 publications

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“…Our work relies heavily on the work of [6], [7] and of other authors so in this section we set the notation and we describe known results on expansion graphs needed for the next sections.…”

Section: Results On Expansion Graphs and Matrix Polynomialsmentioning

confidence: 99%

“…In [7] S. Friedland and H. Schneider defined the concept of expansion graph for an unweighted directed graph, which is obtained from the given graph by replacing certain edges by (exactly) one chain; they studied the effect of graph expansions on the spectrum (of the adjacency matrix) of the graph. In [6] we generalized this concept to weighted graphs by replacing certain edges of the given graph by possibly several disjoint chains of different lengths. One essential observation was that the nonzero spectrum of the adjacency matrix of an expansion graph coincides with the nonzero spectrum of a matrix polynomial with nonnegative coefficients [6, Prop.…”

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confidence: 99%

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On spectra of expansion graphs and matrix polynomials, II

Foerster¹,

Nagy²

2002

ELA

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Abstract. An expansion graph of a directed weighted graph G 0 is obtained from G 0 by replacing some edges by disjoint chains. The adjacency matrix of an expansion graph is a partial linearization of a matrix polynomial with nonnegative coefficients. The spectral radii for different expansion graphs of G 0 and correspondingly the spectral radii of matrix polynomials with nonnegative coefficients, which sum up to a fixed matrix, are compared. A limiting formula is proved for the sequence of the spectral radii of a sequence of expansion graphs of G 0 when the lengths of all chains replacing some original edges tend to infinity. It is shown that for all expansion graphs of G 0 the adjacency matrices have the same level characteristic, but they can have different height characteristics as examples show.

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“…Our work relies heavily on the work of [6], [7] and of other authors so in this section we set the notation and we describe known results on expansion graphs needed for the next sections.…”

Section: Results On Expansion Graphs and Matrix Polynomialsmentioning

confidence: 99%

mentioning

confidence: 99%

On spectra of expansion graphs and matrix polynomials, II

Foerster¹,

Nagy²

2002

ELA

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“…The main goal of this paper is to refine these inequalities. Of particular interest is the case where only a single edge is expanded, a situation where the bounds in [6,7] are generally not sharp. We achieve our results through the use of a symbolic dynamic characterization of the spectral radius of an irreducible matrix and using the notion of the elasticity of the Perron root of A with respect to the (i, j)-th entry which is the quantity given by We comment that it is known (see De Kroon, Plaisier, van Groenendael, and Caswell [5] and Caswell [4]) that Thus all our bounds turn out to include a weighted sum over edges of contributions to the spectral radius for the graph expansion (see, for instance, (3.7) and (3.8)).…”

Section: Neumann and N Ormesmentioning

confidence: 99%

“…As such we are able to obtain results about graph expansions as special cases of results on measured graphs. In our new terminology, both [6] and [7] consider lim n→∞ ρ(G, w, α n ) where {α n } is a sequence of length functions on G, where α n → ∞ on some subgraph G ⊂ G. It is just as natural for us to consider the inverse situation, namely, the behavior of lim n→∞ ρ(G, w, α n ), where {α n } is a sequence of length functions tending to zero on a subgraph of G.…”

Section: Neumann and N Ormesmentioning

confidence: 99%

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Bounds for graph expansions via elasticity

Neumann

Ormes

2003

ELA

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Abstract. In two recent papers, one by Friedland and Schneider, the other by Förster and Nagy, the authors used polynomial matrices to study the effect of graph expansions on the spectral radius of the adjacency matrix. Here it is shown that the notion of the elasticity of the entries of a nonnegative matrix coupled with the Variational Principle for Pressure from symbolic dynamics can be used to derive sharper bounds than existing estimates. This is achieved for weighted and unweighted graphs, and the case of equality is characterized. The work is within the framework of studying measured graphs where each edge is assigned a positive length as well as a weight.Key words. Graph expansions, Nonnegative matrices, Elasticity, Matrix polynomials.AMS subject classifications. 05C50, 15A48, 37B10, 92D25. Introduction.In two recent papers Förster and Nagy [6] and Friedland and Schneider [7] consider the effect of an expansion of unweighted and weighted directed graphs, respectively, on the spectral radius of its adjacency matrix. A graph expansion is obtained from a given graph when an edge is replaced by a path. One of the key results is that the spectral radius strictly decreases under such an operation on an unweighted irreducible graph.The Perron-Frobenius theory does not directly apply in this situation since one is comparing spectral radii of adjacency matrices of different sizes. However, as demonstrated in [7] and [6], one can write equal size adjacency matrices for a graph and its expansion if one allows polynomial entries in the matrix. Polynomial matrices have also arisen as important tools for realization and classification problems in symbolic dynamics; see [1,3,8].More formally, let G = (V, E) be an irreducible directed graph with a finite vertex set V and a finite edge set E (we allow for multiple edges between two vertices). Let w : E → R + (for X ⊂ R, we use the notation X + to denote the set of elements of X which are strictly positive). We call the function w a weight function and the pair (G, w) a weighted graph. Let W = (w(i, j)) be the adjacency matrix for (G, w), i.e., w(i, j) is the sum of weights of edges from vertex i to vertex j. Now suppose that for each edge e ∈ E, we assign a positive integer α(e) and modify the graph by the following process: introduce α(e) − 1 new vertices to the graph, remove the edge e and replace e with a path P(e) of length α(e) which originates at the initial vertex of e, passes through the α(e) − 1 new vertices, and ends at the terminal vertex of e. If we assign the weight of w(e) to the first edge in P(e) and 1 to all others, we obtain a new weighted graph which we will denote (G α , w α ).

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On the Block Numerical Range of Nonnegative Matrices

Förster

Hartanto

2008

Spectral Theory in Inner Product Spaces and Applications

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On spectra of expansion graphs and matrix polynomials

Cited by 4 publications

References 13 publications

On spectra of expansion graphs and matrix polynomials, II

On spectra of expansion graphs and matrix polynomials, II

Bounds for graph expansions via elasticity

On the Block Numerical Range of Nonnegative Matrices

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