There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Why doesn't your home page appear on the first page of search results, even when you query your own name? How do other web pages always appear at the top? What creates these powerful rankings? And how? The first book ever about the science of web page rankings, Google's PageRank and Beyond supplies the answers to these and other questions and more. The book serves two very different audiences: the curious science reader and the technical computational reader. The chapters build in mathematical sophistication, so that the first five are accessible to the general academic reader. While other chapters are much more mathematical in nature, each one contains something for both audiences. For example, the authors include entertaining asides such as how search engines make money and how the Great Firewall of China influences research. The book includes an extensive background chapter designed to help readers learn more about the mathematics of search engines, and it contains several MATLAB codes and links to sample web data sets. The philosophy throughout is to encourage readers to experiment with the ideas and algorithms in the text. Any business seriously interested in improving its rankings in the major search engines can benefit from the clear examples, sample code, and list of resources provided. Many illustrative examples and entertaining asides MATLAB code Accessible and informal style Complete and self-contained section for mathematics review

/pdf/google-s-pagerank-and-beyond-the-science-of-search-engine-33ceqwp28g.pdf

Google's PageRank and Beyond: The Science of Search Engine Rankings

Tables of integrals

We consider the set G consisting of graphs of fixed order and weighted edges. The vertex set of graphs in G will correspond to point masses and the weight for an edge between two vertices is a functional of the distance between them. We pose the problem of finding the best vertex positional configuration in the presence of an additional proximity constraint, in the sense that, the second smallest eigenvalue of the corresponding graph Laplacian is maximized. In many recent applications of algebraic graph theory in systems and control, the second smallest eigenvalue of Laplacian has emerged as a critical parameter that influences the stability and robustness properties of dynamic systems that operate over an information network. Our motivation in the present work is to "assign" this Laplacian eigenvalue when relative positions of various elements dictate the interconnection of the underlying weighted graph. In this venue, one would then be able to "synthesize" information graphs that have desirable system theoretic properties.

On maximizing the second smallest eigenvalue of a state-dependent graph Laplacian

The effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights. The effective resistance comes up in many applications and fields in addition to electrical network analysis, including, for example, Markov chains and continuous-time averaging networks. In this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance, i.e., the sum of the resistances between all pairs of nodes. We show that this is a convex optimization problem and can be solved efficiently either numerically or, in some cases, analytically. We show that optimal allocation of the edge weights can reduce the total effective resistance of the graph (compared to uniform weights) by a factor that grows unboundedly with the size of the graph. We show that among all graphs with $n$ nodes, the path has the largest value of optimal total effective resistance and the complete graph has the least.

/pdf/minimizing-effective-resistance-of-a-graph-1rz525m2td.pdf

Minimizing Effective Resistance of a Graph

https://www.math.ucdavis.edu/~saito/data/distmat/bapat-kirkland-neumann-distmat-lap.pdf

On distance matrices and Laplacians

http://www.hamilton.ie/skirkland/NLEnergy-Final.pdf

On the normalized Laplacian energy and general Randić index R-1 of graphs

The main problem of interest is to investigate how the algebraic connectivity o f a weighted connected graph behaves when the graph is perturbed by removing one or more connected components at a xed vertex and replacing this collection by a single connected component. This analysis leads to exhibiting the unique up to isomorphismtrees on n vertices with speciied diameter that maximize and minimize the algebraic connectivity o ver all such trees. When the radius of a graph is the speciied constraint the unique minimizer of the algebraic connectivity o ver all such graphs is also determined. Analogous results are proved for unicyclic graphs with xed girth. In particular, the unique minimizer and maximizer of the algebraic connectivity is given over all such graphs with girth 3.

/pdf/extremizing-algebraic-connectivity-subject-to-graph-a2l1rp7n4g.pdf

Extremizing algebraic connectivity subject to graph theoretic constraints

We consider a weighted tree T with algebraic connectivity μ, and characteristic vertex v. We show that μ and its associated eigenvectors can be described in terms of the Perron value and vector of a nonnegative matrix which can be computed from the branches of T at v. The machinery of Perron-Frobenius theory can then be used to characterize Type I and Type II trees in terms of these Perron values, and to show that if we construct a weighted tree by taking two weighted trees and identifying a vertex of one with a vertex of the other, then any characteristic vertex of the new tree lies on the path joining the characteristic vertices of the two old trees.

Characteristic vertices of weighted trees via perron values

http://www.math.wsu.edu/faculty/tsat/files/BIRSgraph.pdf

Steve Kirkland

Papers

On distance matrices and Laplacians

On the normalized Laplacian energy and general Randić index R-1 of graphs

Extremizing algebraic connectivity subject to graph theoretic constraints

Characteristic vertices of weighted trees via perron values

Skew-adjacency matrices of graphs