9 research outputs found
The semicircle law for semiregular bipartite graphs
AbstractWe give the (Ahumada type) Selberg trace formula for a semiregular bipartite graph G. Furthermore, we discuss the distribution on arguments of poles of zeta functions of semiregular bipartite graphs. As an application, we present two analogs of the semicircle law for the distribution of eigenvalues of specified regular subgraphs of semiregular bipartite graphs
Low-Density Code-Domain NOMA: Better Be Regular
A closed-form analytical expression is derived for the limiting empirical
squared singular value density of a spreading (signature) matrix corresponding
to sparse low-density code-domain (LDCD) non-orthogonal multiple-access (NOMA)
with regular random user-resource allocation. The derivation relies on
associating the spreading matrix with the adjacency matrix of a large
semiregular bipartite graph. For a simple repetition-based sparse spreading
scheme, the result directly follows from a rigorous analysis of spectral
measures of infinite graphs. Turning to random (sparse) binary spreading, we
harness the cavity method from statistical physics, and show that the limiting
spectral density coincides in both cases. Next, we use this density to compute
the normalized input-output mutual information of the underlying vector channel
in the large-system limit. The latter may be interpreted as the achievable
total throughput per dimension with optimum processing in a corresponding
multiple-access channel setting or, alternatively, in a fully-symmetric
broadcast channel setting with full decoding capabilities at each receiver.
Surprisingly, the total throughput of regular LDCD-NOMA is found to be not only
superior to that achieved with irregular user-resource allocation, but also to
the total throughput of dense randomly-spread NOMA, for which optimum
processing is computationally intractable. In contrast, the superior
performance of regular LDCD-NOMA can be potentially achieved with a feasible
message-passing algorithm. This observation may advocate employing regular,
rather than irregular, LDCD-NOMA in 5G cellular physical layer design.Comment: Accepted for publication in the IEEE International Symposium on
Information Theory (ISIT), June 201
Resolvent of Large Random Graphs
We analyze the convergence of the spectrum of large random graphs to the
spectrum of a limit infinite graph. We apply these results to graphs converging
locally to trees and derive a new formula for the Stieljes transform of the
spectral measure of such graphs. We illustrate our results on the uniform
regular graphs, Erdos-Renyi graphs and preferential attachment graphs. We
sketch examples of application for weighted graphs, bipartite graphs and the
uniform spanning tree of n vertices.Comment: 21 pages, 1 figur
Global eigenvalue fluctuations of random biregular bipartite graphs
We compute the eigenvalue fluctuations of uniformly distributed random
biregular bipartite graphs with fixed and growing degrees for a large class of
analytic functions. As a key step in the proof, we obtain a total variation
distance bound for the Poisson approximation of the number of cycles and
cyclically non-backtracking walks in random biregular bipartite graphs, which
might be of independent interest. As an application, we translate the results
to adjacency matrices of uniformly distributed random regular hypergraphs.Comment: 45 pages, 5 figure
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