520 research outputs found
Quadratic embedding constants of graphs: Bounds and distance spectra
The quadratic embedding constant (QEC) of a finite, simple, connected graph
is the maximum of the quadratic form of the distance matrix of on the
subset of the unit sphere orthogonal to the all-ones vector. The study of these
QECs was motivated by the classical work of Schoenberg on quadratic embedding
of metric spaces [Ann. of Math., 1935] and [Trans. Amer. Math. Soc., 1938]. In
this article, we provide sharp upper and lower bounds for the QEC of trees. We
next explore the relation between distance spectra and quadratic embedding
constants of graphs - and show two further results: We show that the
quadratic embedding constant of a graph is zero if and only if its second
largest distance eigenvalue is zero. We identify a new subclass of
nonsingular graphs whose QEC is the second largest distance eigenvalue.
Finally, we show that the QEC of the cluster of an arbitrary graph with
either a complete or star graph can be computed in terms of the QEC of . As
an application of this result, we provide new families of examples of graphs of
QE class.Comment: 15 pages, 2 figure
How Bad is Forming Your Own Opinion?
The question of how people form their opinion has fascinated economists and
sociologists for quite some time. In many of the models, a group of people in a
social network, each holding a numerical opinion, arrive at a shared opinion
through repeated averaging with their neighbors in the network. Motivated by
the observation that consensus is rarely reached in real opinion dynamics, we
study a related sociological model in which individuals' intrinsic beliefs
counterbalance the averaging process and yield a diversity of opinions.
By interpreting the repeated averaging as best-response dynamics in an
underlying game with natural payoffs, and the limit of the process as an
equilibrium, we are able to study the cost of disagreement in these models
relative to a social optimum. We provide a tight bound on the cost at
equilibrium relative to the optimum; our analysis draws a connection between
these agreement models and extremal problems that lead to generalized
eigenvalues. We also consider a natural network design problem in this setting:
which links can we add to the underlying network to reduce the cost of
disagreement at equilibrium
Least Squares Ranking on Graphs
Given a set of alternatives to be ranked, and some pairwise comparison data,
ranking is a least squares computation on a graph. The vertices are the
alternatives, and the edge values comprise the comparison data. The basic idea
is very simple and old: come up with values on vertices such that their
differences match the given edge data. Since an exact match will usually be
impossible, one settles for matching in a least squares sense. This formulation
was first described by Leake in 1976 for rankingfootball teams and appears as
an example in Professor Gilbert Strang's classic linear algebra textbook. If
one is willing to look into the residual a little further, then the problem
really comes alive, as shown effectively by the remarkable recent paper of
Jiang et al. With or without this twist, the humble least squares problem on
graphs has far-reaching connections with many current areas ofresearch. These
connections are to theoretical computer science (spectral graph theory, and
multilevel methods for graph Laplacian systems); numerical analysis (algebraic
multigrid, and finite element exterior calculus); other mathematics (Hodge
decomposition, and random clique complexes); and applications (arbitrage, and
ranking of sports teams). Not all of these connections are explored in this
paper, but many are. The underlying ideas are easy to explain, requiring only
the four fundamental subspaces from elementary linear algebra. One of our aims
is to explain these basic ideas and connections, to get researchers in many
fields interested in this topic. Another aim is to use our numerical
experiments for guidance on selecting methods and exposing the need for further
development.Comment: Added missing references, comparison of linear solvers overhauled,
conclusion section added, some new figures adde
Spectral graph theory and the inverse eigenvalue problem of a graph
Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graph spectra have been studied extensively for more than fifty years. In the last fifteen years, interest has developed in the study of generalized Laplacian matrices of a graph, that is, real symmetric matrices with negative off-diagonal entries in the positions described by the edges of the graph ( and zero in every other off-diagonal position).The set of all real symmetric matrices having nonzero off-diagonal entries exactly where the graph G has edges is denoted by S( G). Given a graph G, the problem of characterizing the possible spectra of B, such that B. S( G), has been referred to as the Inverse Eigenvalue Problem of a Graph. In the last fifteen years a number of papers on this problem have appeared, primarily concerning trees.The adjacency matrix and Laplacian matrix of G and their normalized forms are all in S( G). Recent work on generalized Laplacians and Colin de Verdiere matrices is bringing the two areas closer together. This paper surveys results in Spectral Graph Theory and the Inverse Eigenvalue Problem of a Graph, examines the connections between these problems, and presents some new results on construction of a matrix of minimum rank for a given graph having a special form such as a 0,1-matrix or a generalized Laplacian
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