224 research outputs found
Cyclotomic matrices over real quadratic integer rings
We classify all cyclotomic matrices over real quadratic integer rings and we
show that this classification is the same as classifying cyclotomic matrices
over the compositum all real quadratic integer rings. Moreover, we enumerate a
related class of symmetric matrices; those matrices whose eigenvalues are
contained inside the interval [-2,2] but whose characteristic polynomials are
not in Z[x].Comment: 13 page
Ramanujan Coverings of Graphs
Let be a finite connected graph, and let be the spectral radius of
its universal cover. For example, if is -regular then
. We show that for every , there is an -covering
(a.k.a. an -lift) of where all the new eigenvalues are bounded from
above by . It follows that a bipartite Ramanujan graph has a Ramanujan
-covering for every . This generalizes the case due to Marcus,
Spielman and Srivastava (2013).
Every -covering of corresponds to a labeling of the edges of by
elements of the symmetric group . We generalize this notion to labeling
the edges by elements of various groups and present a broader scenario where
Ramanujan coverings are guaranteed to exist.
In particular, this shows the existence of richer families of bipartite
Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava,
a crucial component of our proof is the existence of interlacing families of
polynomials for complex reflection groups. The core argument of this component
is taken from a recent paper of them (2015).
Another important ingredient of our proof is a new generalization of the
matching polynomial of a graph. We define the -th matching polynomial of
to be the average matching polynomial of all -coverings of . We show this
polynomial shares many properties with the original matching polynomial. For
example, it is real rooted with all its roots inside .Comment: 38 pages, 4 figures, journal version (minor changes from previous
arXiv version). Shortened version appeared in STOC 201
Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings
We solve two inverse spectral problems for star graphs of Stieltjes strings
with Dirichlet and Neumann boundary conditions, respectively, at a selected
vertex called root. The root is either the central vertex or, in the more
challenging problem, a pendant vertex of the star graph. At all other pendant
vertices Dirichlet conditions are imposed; at the central vertex, at which a
mass may be placed, continuity and Kirchhoff conditions are assumed. We derive
conditions on two sets of real numbers to be the spectra of the above Dirichlet
and Neumann problems. Our solution for the inverse problems is constructive: we
establish algorithms to recover the mass distribution on the star graph (i.e.
the point masses and lengths of subintervals between them) from these two
spectra and from the lengths of the separate strings. If the root is a pendant
vertex, the two spectra uniquely determine the parameters on the main string
(i.e. the string incident to the root) if the length of the main string is
known. The mass distribution on the other edges need not be unique; the reason
for this is the non-uniqueness caused by the non-strict interlacing of the
given data in the case when the root is the central vertex. Finally, we relate
of our results to tree-patterned matrix inverse problems.Comment: 32 pages, 3 figure
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