1,994 research outputs found
The extremal spectral radii of -uniform supertrees
In this paper, we study some extremal problems of three kinds of spectral
radii of -uniform hypergraphs (the adjacency spectral radius, the signless
Laplacian spectral radius and the incidence -spectral radius).
We call a connected and acyclic -uniform hypergraph a supertree. We
introduce the operation of "moving edges" for hypergraphs, together with the
two special cases of this operation: the edge-releasing operation and the total
grafting operation. By studying the perturbation of these kinds of spectral
radii of hypergraphs under these operations, we prove that for all these three
kinds of spectral radii, the hyperstar attains uniquely the
maximum spectral radius among all -uniform supertrees on vertices. We
also determine the unique -uniform supertree on vertices with the second
largest spectral radius (for these three kinds of spectral radii). We also
prove that for all these three kinds of spectral radii, the loose path
attains uniquely the minimum spectral radius among all
-th power hypertrees of vertices. Some bounds on the incidence
-spectral radius are given. The relation between the incidence -spectral
radius and the spectral radius of the matrix product of the incidence matrix
and its transpose is discussed
Numerical Algorithms for Polynomial Optimisation Problems with Applications
In this thesis, we study tensor eigenvalue problems and polynomial optimization problems. In particular, we present a fast algorithm for computing the spectral radii of symmetric nonnegative tensors without requiring the partition of the tensors. We also propose some polynomial time approximation algorithms with new approximation bounds for nonnegative polynomial optimization problems over unit spheres. Furthermore, we develop an efficient and effective algorithm for the maximum clique problem
Perron communicability and sensitivity of multilayer networks
Modeling complex systems that consist of different types of objects leads to
multilayer networks, where nodes in the different layers represent different
kind of objects. Nodes are connected by edges, which have positive weights. A
multilayer network is associated with a supra-adjacency matrix. This paper
investigates the sensitivity of the communicability in a multilayer network to
perturbations of the network by studying the sensitivity of the Perron root of
the supra-adjacency matrix. Our analysis sheds light on which edge weights to
make larger to increase the communicability of the network, and which edge
weights can be made smaller or set to zero without affecting the
communicability significantly.Comment: 20 pages, 1 figure, 7 table
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