Computing the Characteristic polynomial of a graph

Schwenk, Allen J.

doi:10.1007/bfb0066438

Cited by 265 publications

(150 citation statements)

References 7 publications

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“…For x-Rössler systems in a k-cycle, this cost tends to f = 2k min /(n − 1) ≃ 0.140. Other constructive lattices [22,24] also tend to constant fractions: f = 0.252 (for k-wheels), f = 0.070 (for k-Möbius ladders), and f = 0.053 (for the most economical bipartite graph). In all those cases, the cost of synchronization is high: the necessary number of edges scales like ∼ n 2 , just like the complete graph.…”

Section: Pacs Numbersmentioning

confidence: 99%

Synchronization in Small-World Systems

2002

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We quantify the dynamical implications of the small-world phenomenon. We consider the generic synchronization of oscillator networks of arbitrary topology, and link the linear stability of the synchronous state to an algebraic condition of the Laplacian of the graph. We show numerically that the addition of random shortcuts produces improved network synchronizability. Further, we use a perturbation analysis to place the synchronization threshold in relation to the boundaries of the small-world region. Our results also show that small-worlds synchronize as efficiently as random graphs and hypercubes, and more so than standard constructive graphs. PACS numbers:Recently, Watts and Strogatz [1] showed that the addition of a few long-range shortcuts to an otherwise locally connected lattice (the "pristine world") produces a sharp reduction of the average distance between arbitrary nodes. The ensuing semi-random lattice was denoted a small-world (SW) because the sudden appearance of short paths occurs early on, while the system is still relatively localized. This concept has wide appeal: the SW property seems to be a quantifiable characteristic of many real-world structures [1, 2, 3, 4], both human generated (social networks, WWW, power grid), or of biological origin (neural and biochemical networks).A spur of ongoing research [2] has concentrated on static and combinatoric properties [5,6,7,8,9, 10] of a tractable SW model [1,11]. Monasson [11] considered the SW effect on the distribution of eigenvalues of the connectivity matrix (the graph Laplacian) which specifies the coupling between nodes-a relevant topic for polymer networks [12]. However, despite their central role in real-world networks, there are fewer studies of dynamical processes taking place on SW lattices. Among those, automata epidemics simulations [13] In this paper, we explicitly link the SW addition of random shortcuts to the synchronization of networks of coupled dynamical systems. This is an example of dynamics on networks-leaving aside the distinct problem of evolution of networks here. By using a generic synchronization formulation [17,18] to factor out the connectivity of the network, we identify the synchronization threshold with an algebraic condition of the graph Laplacian. Through numerics and analysis, we quantify how the SW scheme improves the synchronizability of the pristine world, mainly as a result of the steep increase of the first-non-zero eigenvalue (FNZE). The synchronization threshold is found to lie in the SW region [3, 13], but does not coincide with its onset-it can in fact be linked to the effective randomization that ends SW. Within this framework, we show that the synchronization efficiency of semi-random SW networks is higher than standard deterministic graphs, and comparable to both fully random and ideal constructive graphs.Consider n identical dynamical systems (placed at the nodes of a graph) that are linearly and symmetrically coupled (as represented by the edges of the undirected graph) with global coupling strengt...

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Section: Pacs Numbersmentioning

confidence: 99%

Synchronization in Small-World Systems

2002

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“…One important property of covers discussed in Section 3.2 is that the spectrum of any graph embeds (as a multiset, i.e., taking into account also the multiplicities of the eigenvalues) in the spectrum of its cover. This result can be proven in many ways, for example as a consequence of either Theorem 0.12 or Theorem 4.7, both of [8], or using the notion of equitable partitions introduced by Schwenk [39]. (See [32] for a proof based on the latter approach.)…”

Section: Theorem 3 For Every a ⊆ R And Everymentioning

confidence: 99%

General properties of some families of graphs defined by systems of equations

Lazebnik

Woldar

2001

Journal of Graph Theory

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Abstract. In this paper we present a simple method for constructing infinite families of graphs defined by a class of systems of equations over commutative rings. We show that the graphs in all such families possess some general properties including regularity and bi-regularity, existence of special vertex colorings, and existence of covering maps -hence, embedded spectra -between every two members of the same family. Another general property, recently discovered, is that nearly every graph constructed in this manner edge-decomposes either the complete, or complete bipartite, graph which it spans.In many instances, specializations of these constructions have proved useful in various graph theory problems, but especially in many extremal problems. A short survey of the related results is included. We also show that the edgedecomposition property allows one to improve existing lower bounds for some multicolor Ramsey numbers.

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“…An analogous result for graphs, namely that the characteristic polynomial of S divides the characteristic polynomial of a graph was obtained by Mowshowitz [10], and by Schwenk [15]. …”

Section: Equitable Partition For Edmsmentioning

confidence: 54%

“…Schwenk [15] used equitable partitions to find the eigenvalues of the adjacency matrix of a graph. Hayden et al [7] also used equitable partitions, albeit under the name block structure, to investigate EDMs generated by points lying on a collection of concentric spheres.…”

Section: Introductionmentioning

confidence: 99%

On the eigenvalues of Euclidean distance matrices

Alfakih¹

2008

Comput.appl.math.

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Abstract. In this paper, the notion of equitable partitions (EP) is used to study the eigenvalues of Euclidean distance matrices (EDMs). In particular, EP is used to obtain the characteristic polynomials of regular EDMs and non-spherical centrally symmetric EDMs. The paper also presents methods for constructing cospectral EDMs and EDMs with exactly three distinct eigenvalues.Mathematical subject classification: 51K05, 15A18, 05C50.

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Computing the Characteristic polynomial of a graph

Cited by 265 publications

References 7 publications

Synchronization in Small-World Systems

Synchronization in Small-World Systems

General properties of some families of graphs defined by systems of equations

On the eigenvalues of Euclidean distance matrices

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