Eigenvalues of Totally Positive Integral Operators
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It is known [10, 11] that if T is an integral operator with an extended totally positive kernel, then T has a countably infinite family of simple, positive eigenvalues. We prove a similar result for a rather larger class of kernels and, writing the eigenvalues of T in decreasing order as (λn)n∈N, we use results obtained in [4] and [5] to give a formula for the ratio λn+1/λn analogous to that given in [3] for the case of a strictly totally positive matrix, and to the spectral radius formula r ( T ) = lim n → ∞ ‖ T n ‖ 1 / n = inf n ∈ N ‖ T n ‖ 1 / n . This may be regarded as a generalisation of inequalities due to Hopf [8, 9]. 1991 1991 Mathematics Subject Classification 47G10, 47B65.Keywords:
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Matrix (chemical analysis)
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The distance spectral radius of a connected hypergraph is the largest eigenvalue of its distance matrix. We determine the unique connected k-uniform hypergraphs with minimum distance spectral radius when the number of pendant edges is given, the unique k-uniform non-hyperstar-like hypertrees (non-hyper-caterpillars, respectively) with minimum distance spectral radius, and the unique k-uniform non-hyper-caterpillars with maximum distance spectral radius for .
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In 1970 Smith classified all connected graphs with the spectral radius at most $2$. Here the spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Recently, the definition of spectral radius has been extended to $r$-uniform hypergraphs. In this paper, we generalize the Smith's theorem to $r$-uniform hypergraphs. We show that the smallest limit point of the spectral radii of connected $r$-uniform hypergraphs is $\rho_r=(r-1)!\sqrt[r]{4}$. We discovered a novel method for computing the spectral radius of hypergraphs, and classified all connected $r$-uniform hypergraphs with spectral radius at most $\rho_r$.
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