Calculating two-point resistances in distance-regular resistor networks

Jafarizadeh, M. A.; Sufiani, R.; Jafarizadeh, Saber

doi:10.1088/1751-8113/40/19/002

Cited by 34 publications

(61 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…See also [56,15]. In this section, we consider only simple graphs; the results can be extended to multigraphs but statements are more elaborate.…”

Section: Foster's Formulae and Their Generalizationmentioning

confidence: 99%

Combinatorics of optimal designs

Bailey¹,

Cameron²

2009

Surveys in Combinatorics 2009

View full text Add to dashboard Cite

To a combinatorialist, a design is usually a 2-design or balanced incompleteblock design. However, 2-designs do not necessarily exist in all cases where a statistician might wish to use one to design an experiment. As a result, statisticians need to consider structures much more general than the combinatorialist's designs, and to decide which one is "best" in a given situation. This leads to the theory of optimal design. There are several concepts of optimality, and no general consensus about which one to use in any particular situation.For block designs with fixed block size k, all these optimality criteria are determined by a graph, the concurrence graph of the design, and more specifically, by the eigenvalues of the Laplacian matrix of the graph. It turns out that the optimality criteria most used by statisticians correspond to properties of this graph which are interesting in other contexts: D-optimality involves maximizing the number of spanning trees; A-optimality involves minimizing the sum of resistances between all pairs of terminals (when the graph is regarded as an electrical circuit, with each edge being a one-ohm resistor); and E-optimality involves maximizing the smallest eigenvalue of the Laplacian (the corresponding graphs are likely to have good expansion and random walk properties). If you are familiar with these properties, you may expect that related "nice" properties such as regularity and large girth (or even symmetry) may tend to hold; some of our examples may come as a surprise! The aim of this paper is to point out that the optimal design point of view unifies various topics in graph theory and design theory, and suggests some interesting open problems to which combinatorialists of all kinds might turn their expertise. We describe in some detail both the statistical background and the mathematics of various topics such as Laplace eigenvalues of graphs. PreliminariesThis first section of the paper sets the scene. We look briefly at the different ways in which a combinatorialist and a statistician view a simple block design such as the Fano plane, and also introduce a running example which would not be recognised as a design under the standard combinatorial definition but is in fact the best design for a statistician in certain circumstances.Section 2 looks at the way that information about treatment differences is recovered from the results of an experiment conducted using a block design. We look briefly at what makes a good design, and show that all criteria for this can be expressed in terms of the concurrence graph of the design. Having thus focussed our attention on graphs, we give a brief survey of the Laplacian matrix of a graph and its eigenvalues, and mention the interesting question of which graphs can be concurrence graphs of block designs with given block size.Section 3 covers the three most important kinds of optimality, described by the letters A (average), D (determinant) and E (extreme), and how they look when expressed in terms of the concurrence matrix.In section ...

show abstract

“…See also [56,15]. In this section, we consider only simple graphs; the results can be extended to multigraphs but statements are more elaborate.…”

Section: Foster's Formulae and Their Generalizationmentioning

confidence: 99%

Combinatorics of optimal designs

Bailey¹,

Cameron²

2009

Surveys in Combinatorics 2009

View full text Add to dashboard Cite

show abstract

“…It should be noticed that, by using the above result, one can evaluate the limiting value of the effective resistances in the limit of the large size of the networks, i.e., in the limit N → ∞. For instance, the effective resistances R αβ (1) , R αβ (2) and R αβ (3) tend to…”

Section: Examplesmentioning

confidence: 91%

“…respectively (for more details see [1]). The intersection numbers (2-7) and the valencies κ i satisfy the following obvious conditions…”

Section: Furthermore If We Havementioning

confidence: 99%

“…Recently, in Refs. [1] and [2], calculation of effective resistances on distance-regular networks was investigated, where in the first paper, the calculation was based on stratification and Stieltjes function associated with the network, whereas in the latter one a recursive formula for effective resistances was given based on the Christoffel-Darboux identity for orthogonal polynomials. In this paper, we consider more general resistor networks which are underlying networks of association schemes.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Calculating effective resistances on underlying networks of association schemes

Jafarizadeh

Sufiani

Jafarizadeh

2008

Journal of Mathematical Physics

View full text Add to dashboard Cite

Recently, in Refs. [1] and [2], calculation of effective resistances on distance-regular networks was investigated, where in the first paper, the calculation was based on stratification and Stieltjes function associated with the network, whereas in the latter one a recursive formula for effective resistances was given based on the Christoffel-Darboux identity. In this paper, evaluation of effective resistances on more general networks which are underlying networks of association schemes is considered, where by using the algebraic combinatoric structures of association schemes such as stratification and Bose-Mesner algebras, an explicit formula for effective resistances on these networks is given in terms of the parameters of corresponding association schemes. Moreover, we show that for particular underlying networks of association schemes with diameter d such that the adjacency matrix A possesses d + 1 distinct eigenvalues, all of the other adjacency matrices A i , i = 0, 1 can be written as polynomials of A, i.e., A i = P i (A), where P i is not necessarily of degree i. Then, we use this property for these particular networks and assume that all of the conductances except for one of them, say c ≡ c 1 = 1, are zero to give a procedure for evaluating effective resistances on these networks. The preference of this procedure is that one can evaluate effective resistances by using the structure of their Bose-Mesner algebra without any need to know the spectrum of the adjacency matrices.

show abstract

“…This is a classical problem of multilinear algebra which has been recently solved by Wu [19] in an elegant manner using the formalism of dyadic algebra or, in a more abstract form, by Jafarizadeh et al [20] whose calculation is based on stratification of the underlying graph and the Stieltjes transform of the spectral distribution associated with the network.…”

Section: Resistance Between Two Diagonally Opposite Nodesmentioning

confidence: 99%

Electrical resistance between pairs of vertices of a conducting cube and continuum limit for a cubic resistor network

Mamode

2017

J. Phys. Commun.

View full text Add to dashboard Cite

A method is given of determining analytically the driving-point resistance between any pair of vertices of: (i) a homogeneous conducting cube, (ii) a N N Ń´cubic lattice of identical resistors. For the latter, the limit N  ¥ is considered and the condition of the ohmic equivalence between these two systems is investigated.-, the resistance may be in principle obtained as the ratio OPEN ACCESS RECEIVED

show abstract

Calculating two-point resistances in distance-regular resistor networks

Cited by 34 publications

References 38 publications

Combinatorics of optimal designs

Combinatorics of optimal designs

Calculating effective resistances on underlying networks of association schemes

Electrical resistance between pairs of vertices of a conducting cube and continuum limit for a cubic resistor network

Contact Info

Product

Resources

About