20,910 research outputs found
Extrema of graph eigenvalues
In 1993 Hong asked what are the best bounds on the 'th largest eigenvalue
of a graph of order . This challenging question has
never been tackled for any . In the present paper tight bounds are
obtained for all and even tighter bounds are obtained for the 'th
largest singular value
Some of these bounds are based on Taylor's strongly regular graphs, and other
on a method of Kharaghani for constructing Hadamard matrices. The same kind of
constructions are applied to other open problems, like Nordhaus-Gaddum problems
of the kind: How large can be
These constructions are successful also in another open question: How large
can the Ky Fan norm be
Ky Fan norms of graphs generalize the concept of graph energy, so this question
generalizes the problem for maximum energy graphs.
In the final section, several results and problems are restated for
-matrices, which seem to provide a more natural ground for such
research than graphs.
Many of the results in the paper are paired with open questions and problems
for further study.Comment: 32 page
Interlacing Families IV: Bipartite Ramanujan Graphs of All Sizes
We prove that there exist bipartite Ramanujan graphs of every degree and
every number of vertices. The proof is based on analyzing the expected
characteristic polynomial of a union of random perfect matchings, and involves
three ingredients: (1) a formula for the expected characteristic polynomial of
the sum of a regular graph with a random permutation of another regular graph,
(2) a proof that this expected polynomial is real rooted and that the family of
polynomials considered in this sum is an interlacing family, and (3) strong
bounds on the roots of the expected characteristic polynomial of a union of
random perfect matchings, established using the framework of finite free
convolutions we recently introduced
An Efficient Parallel Solver for SDD Linear Systems
We present the first parallel algorithm for solving systems of linear
equations in symmetric, diagonally dominant (SDD) matrices that runs in
polylogarithmic time and nearly-linear work. The heart of our algorithm is a
construction of a sparse approximate inverse chain for the input matrix: a
sequence of sparse matrices whose product approximates its inverse. Whereas
other fast algorithms for solving systems of equations in SDD matrices exploit
low-stretch spanning trees, our algorithm only requires spectral graph
sparsifiers
Beyond graph energy: norms of graphs and matrices
In 1978 Gutman introduced the energy of a graph as the sum of the absolute
values of graph eigenvalues, and ever since then graph energy has been
intensively studied.
Since graph energy is the trace norm of the adjacency matrix, matrix norms
provide a natural background for its study. Thus, this paper surveys research
on matrix norms that aims to expand and advance the study of graph energy.
The focus is exclusively on the Ky Fan and the Schatten norms, both
generalizing and enriching the trace norm. As it turns out, the study of
extremal properties of these norms leads to numerous analytic problems with
deep roots in combinatorics.
The survey brings to the fore the exceptional role of Hadamard matrices,
conference matrices, and conference graphs in matrix norms. In addition, a vast
new matrix class is studied, a relaxation of symmetric Hadamard matrices.
The survey presents solutions to just a fraction of a larger body of similar
problems bonding analysis to combinatorics. Thus, open problems and questions
are raised to outline topics for further investigation.Comment: 54 pages. V2 fixes many typos, and gives some new materia
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