109 research outputs found
Generalized Sharp Bounds on the Spectral Radius of Digraphs
The spectral radius {\rho}(G) of a digraph G is the maximum modulus of the
eigenvalues of its adjacency matrix. We present bounds on {\rho}(G) that are
often tighter and are applicable to a larger class of digraphs than previously
reported bounds. Calculating the final bound pair is particularly suited to
sparse digraphs.
For strongly connected digraphs, we derive equality conditions for the
bounds, relating to the outdegree regularity of the digraph. We also prove that
the bounds hold with equality only if {\rho}(G) is the r-th root of an integer,
where r divides the index of imprimitivity of G.Comment: 12 pages and 1 figure. Nov. 30, 2012 revisio
Dynamics over Signed Networks
A signed network is a network with each link associated with a positive or
negative sign. Models for nodes interacting over such signed networks, where
two different types of interactions take place along the positive and negative
links, respectively, arise from various biological, social, political, and
economic systems. As modifications to the conventional DeGroot dynamics for
positive links, two basic types of negative interactions along negative links,
namely the opposing rule and the repelling rule, have been proposed and studied
in the literature. This paper reviews a few fundamental convergence results for
such dynamics over deterministic or random signed networks under a unified
algebraic-graphical method. We show that a systematic tool of studying node
state evolution over signed networks can be obtained utilizing generalized
Perron-Frobenius theory, graph theory, and elementary algebraic recursions.Comment: In press, SIAM Revie
Totally Nonnegative (0, 1)-Matrices
We investigate (0, 1)-matrices which are totally nonnegative and therefore which have all of their eigenvalues equal to nonnegative real numbers. Such matrices are characterized by four forbidden submatrices (of orders 2 and 3). We show that the maximum number of 0s in an irreducible (0, 1)-matrix of order n is (n − 1)2 and
characterize those matrices with this number of 0s. We also show that the minimum Perron value of an irreducible, totally nonnegative (0, 1)-matrix of order n equals 2 + 2 cos (2∏/n+2) and characterize those matrices with this Perron value
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