455 research outputs found
The skew energy of random oriented graphs
Given a graph , let be an oriented graph of with the
orientation and skew-adjacency matrix . The skew energy
of the oriented graph , denoted by , is
defined as the sum of the absolute values of all the eigenvalues of
. In this paper, we study the skew energy of random oriented
graphs and formulate an exact estimate of the skew energy for almost all
oriented graphs by generalizing Wigner's semicircle law. Moreover, we consider
the skew energy of random regular oriented graphs , and get an
exact estimate of the skew energy for almost all regular oriented graphs.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1011.6646 by
other author
Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits
We deploy algebraic complexity theoretic techniques for constructing
symmetric determinantal representations of for00504925mulas and weakly skew
circuits. Our representations produce matrices of much smaller dimensions than
those given in the convex geometry literature when applied to polynomials
having a concise representation (as a sum of monomials, or more generally as an
arithmetic formula or a weakly skew circuit). These representations are valid
in any field of characteristic different from 2. In characteristic 2 we are led
to an almost complete solution to a question of B\"urgisser on the
VNP-completeness of the partial permanent. In particular, we show that the
partial permanent cannot be VNP-complete in a finite field of characteristic 2
unless the polynomial hierarchy collapses.Comment: To appear in the AMS Contemporary Mathematics volume on
Randomization, Relaxation, and Complexity in Polynomial Equation Solving,
edited by Gurvits, Pebay, Rojas and Thompso
Complex spherical codes with three inner products
Let be a finite set in a complex sphere of dimension. Let be
the set of usual inner products of two distinct vectors in . A set is
called a complex spherical -code if the cardinality of is and
contains an imaginary number. We would like to classify the largest
possible -codes for given dimension . In this paper, we consider the
problem for the case . Roy and Suda (2014) gave a certain upper bound for
the cardinalities of -codes. A -code is said to be tight if
attains the bound. We show that there exists no tight -code except for
dimensions , . Moreover we make an algorithm to classify the largest
-codes by considering representations of oriented graphs. By this algorithm,
the largest -codes are classified for dimensions , , with a
current computer.Comment: 26 pages, no figur
- …