Extensions and applications of equitable decompositions for graphs with symmetries

Francis, Amanda; Smith, Dallas; Sorensen, Derek; Webb, Benjamin

doi:10.1016/j.laa.2017.06.045

Cited by 11 publications

(26 citation statements)

References 12 publications

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“…Last, we demonstrate that for any automorphism compatible matrix M and automorphism φ the divisor matrix M φ and the matrix M have the same spectral radius if M is both nonnegative and irreducible (see Proposition 5.1). As this holds for any automorphism φ this extends the result of [2] in which this was shown to hold for both basic and separable automorphisms.…”

Section: Introductionsupporting

confidence: 81%

“…Here, as in [1] and [2] we consider those matrices M associated with a graph G whose structure mimics the symmetries of the graph.…”

Section: Graph Symmetries and Equitable Decompositionsmentioning

confidence: 99%

“…A basic automorphism φ is an automorphism for which all orbits of size greater than one have the same cardinality (see [1]). A separable automorphism is an automorphism whose order is the product of distinct primes (see [2]).…”

Section: Introductionmentioning

confidence: 99%

“…From a practical point of view, the large size of these networks limit our ability to quickly compute their associated eigenvalues and eigenvectors, which are used in a number of standard network metrics and algorithms [7]. However, their high degree of symmetry suggests that it may be possible to effectively estimate a network's spectral properties by equitably decomposing the network over local symmetries, which is a potentially much more feasible task (see Examples 5.3 and 5.4 from [2]). This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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General equitable decompositions for graphs with symmetries

Francis

Smith

Webb

2019

Linear Algebra and its Applications

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Using the theory of equitable decompositions it is possible to decompose a matrix M appropriately associated with a given graph. The result is a collection of smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix M. This is done by decomposing the matrix over a graph symmetry. Previously it was shown that a matrix can be equitably decomposed over any uniform, basic, or separable automorphism. Here we extend this theory to show that it is possible to equitably decompose a matrix over any automorphism of a graph, without restriction. Moreover, we give a step-by-step procedure which can be used to generate such a decomposition. We also prove under mild conditions that if a matrix M is equitably decomposed the resulting divisor matrix, which is the divisor matrix of the associated equitable partition, will have the same spectral radius as the original matrix M.

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Section: Introductionsupporting

confidence: 81%

“…Here, as in [1] and [2] we consider those matrices M associated with a graph G whose structure mimics the symmetries of the graph.…”

Section: Graph Symmetries and Equitable Decompositionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

General equitable decompositions for graphs with symmetries

Francis

Smith

Webb

2019

Linear Algebra and its Applications

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“…By means of the 'equitable partition theorem' [26,46,47] its N + 4 eigenstates can then be shown to be can then be evaluated as…”

Section: Brief Conclusionmentioning

confidence: 99%

Designing pretty good state transfer via isospectral reductions

et al. 2020

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We present an algorithm to design networks that feature pretty good state transfer (PGST), which is of interest for high-fidelity transfer of information in quantum computing. Realizations of PGST networks have so far mostly relied either on very special network geometries or imposed conditions such as transcendental on-site potentials. However, it was recently shown [Eisenberg et al., arXiv:1804.01645] that PGST generally arises when a network's eigenvectors and the factors P± of its characteristic polynomial P fulfill certain conditions, where P± correspond to eigenvectors which have ±1 parity on the input and target sites. We combine this result with the so-called isospectral reduction of a network to obtain P± from a dimensionally reduced form of the Hamiltonian. Equipped with the knowledge of the factors P±, we show how a variety of setups can be equipped with PGST by proper tuning of P±. Having demonstrated a method of designing networks featuring pretty good state transfer of single site excitations, we further show how the obtained networks can be manipulated such that they allow for robust storage of qubits. We hereby rely on the concept of compact localized states, which are eigenstates of a Hamiltonian localized on a small subdomain, and whose amplitudes completely vanish outside of this domain. Such states are natural candidates for the storage of quantum information, and we show how certain Hamiltonians featuring pretty good state transfer of single site excitation can be equipped with compact localized states such that their transfer is made possible.

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Characteristic polynomials and zeta functions of equitably partitioned graphs

Kada

2020

Linear Algebra and its Applications

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Let π = {V 1 , . . . , V r } be an equitable partition of the vertex set of a directed graph (digraph) X. It is well known that the characteristic polynomial φ(X/π, x) of a quotient graph X/π divides that of X, but the remainder part is not well investigated. In this paper, we define a deletion graph X\π over an equitable partition π, which is a signed directed graph defined for a fixed set of deleting vertices {v i ∈ V i , i = 1, · · · , r}, and give a similarity transformation exchanging the adjacency matrix A(X) which is compatible with the equitable partition for a block triangular matrix whose diagonal blocks are the adjacency matrix of the quotient graph and the deletion graph. In fact, we show the result for more general matrices including adjacency matrix of graphs, and as corollaries, we show the followings: (i) a decomposition formula of the reciprocal of the Ihara-Bartholdi zeta function over an equitably partitioned undirected graph into the quotient graph part and the deletion graph part, and (ii) Chen and Chen's result ([CC17, Theorem 3.1]) on the Ihara-Bartholdi zeta functions on generalized join graphs, and (iii) Teranishi's result [Ter03, Theorem 3.3].

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Extensions and applications of equitable decompositions for graphs with symmetries

Cited by 11 publications

References 12 publications

General equitable decompositions for graphs with symmetries

General equitable decompositions for graphs with symmetries

Designing pretty good state transfer via isospectral reductions

Characteristic polynomials and zeta functions of equitably partitioned graphs

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