Distance-Regular Graphs

Zeros of orthogonal polynomials

Preface 1. Introduction 2. Graph operations and modifications 3. Spectrum and structure 4. Characterizations by spectra 5. Structure and one eigenvalue 6. Spectral techniques 7. Laplacians 8. Additional topics 9. Applications Appendix Bibliography Index of symbols Index.

An Introduction to the Theory of Graph Spectra: Spectral techniques

Introduction to coding theory (2nd edition), by J.H. van Lint. Pp.183 DM86. 1992. ISBN 0-387-54894-7 (Springer)

Covering a network with the minimum possible number of boxes can reveal interesting features for the network structure, especially in terms of self-similar or fractal characteristics. Considerable attention has been recently devoted to this problem, with the finding that many real networks are self-similar fractals. Here we present, compare and study in detail a number of algorithms that we have used in previous papers towards this goal. We show that this problem can be mapped to the well-known graph colouring problem and then we simply can apply well-established algorithms. This seems to be the most efficient method, but we also present two other algorithms based on burning which provide a number of other benefits. We argue that the algorithms presented provide a solution close to optimal and that another algorithm that can significantly improve this result in an efficient way does not exist. We offer to anyone that finds such a method to cover his/her expenses for a one-week trip to our lab in New York (details in http://jamlab.org).

/pdf/how-to-calculate-the-fractal-dimension-of-a-complex-network-3wceb96n1j.pdf

How to calculate the fractal dimension of a complex network: the box covering algorithm

https://upcommons.upc.edu/bitstream/handle/2117/672/hierarchical(7feb).pdf;sequence=1

The hierarchical product of graphs

https://upcommons.upc.edu/bitstream/handle/2117/1927/Gen_hierarchical(27mar).pdf;sequence=1

The generalized hierarchical product of graphs

/pdf/on-almost-distance-regular-graphs-4gpdcq3erl.pdf

On almost distance-regular graphs

Distance-regular graphs have been a key concept in Algebraic Combinatorics and have given place to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we study `almost distance-regular graphs'. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. For example, a known characterization of a distance-regular graph is the invariance of the number of walks of given length between vertices at a given distance, while a graph is called walk-regular if the number of closed walks of given length rooted at any given vertex is a constant. One of the concepts studied here is a generalization of both distance-regularity and walk-regularity called m-walk-regularity. Another studied concept is that of m-partial distance-regularity, or informally, distance-regularity up to distance m. Using eigen- values of graphs and the predistance polynomials, we discuss and relate these and other concepts of almost distance-regularity, such as their common generalization of (l, m)-walk-regularity. We introduce the concepts of punctual distance-regularity and punctual walk-regularity as a fundament upon which almost distance-regular graphs are built. We provide examples that are mostly taken from the Foster census, a collection of symmetric cubic graphs. Two problems are posed that are related to the question of when almost distance-regular becomes whole distance-regular. We also give several characterizations of punctually distance-regular graphs that are generalizations of the spectral excess theorem.

On Almost Distance-Regular Graphs

Although some real networks exhibit self-similarity, there is no standard definition of fractality in graphs. On the other hand, the small-world phenomenon is one of the most important common properties of real interconnection networks. In this paper we relate these two properties. In order to do so, we focus on the family of Sierpinski networks. For the Sierpinski gasket, the Sierpinski carpet and the Sierpinski tetra, we give the basic properties and we calculate the box-counting dimension as a measure of their fractality. We also define a deterministic family of graphs, which we call small-world Sierpinski graphs. We show that our construction preserves the structure of Sierpinski graphs, including its box-counting dimension, while the small-world phenomenon arises. Thus, in this family of graphs, fractality and small-world effect are simultaneously present.

https://iopscience.iop.org/article/10.1088/0305-4470/39/38/003/pdf

Cristina Dalfó

Papers

The hierarchical product of graphs

The generalized hierarchical product of graphs

On almost distance-regular graphs

On Almost Distance-Regular Graphs

Fractality and the small-world effect in Sierpinski graphs