On eigenvalues of Laplacian matrix for a class of directed signed graphs

Abstract

The eigenvalues of the Laplacian matrix for a class of directed graphs with both positive and negative weights are studied. First, a class of directed signed graphs is investigated in which one pair of nodes (either connected or not) is perturbed with negative weights. A necessary condition is proposed to attain the following objective for the perturbed graph: the real parts of the non-zero eigenvalues of its Laplacian matrix are positive. A sufficient condition is also presented that ensures the aforementioned objective for unperturbed graph. It is then highlighted the case where the condition becomes necessary and sufficient. Secondly, for directed graphs, a subset of pairs of nodes are identified where if any of the pairs is connected by an edge with infinitesimal negative weight, the resulting Laplacian matrix will have at least one eigenvalue with negative real part. Illustrative examples are presented to show the applicability of our results.

Investigating stability of Laplacians on signed digraphs via eventual positivity

“…Furthermore, from Lemma 6.3 of [19], if L s is psd then ker(L) = ker(L T ) ⊆ ker(L s ). Since ker(L s ) = span (1) and corank(L s ) = 1, and since weight balance of L implies L1 = L T 1 = 0, the implication follows.…”

“…Theorem 4 Consider a strongly connected signed digraph G(A), and the corresponding Laplacian (2). If −L is eventually exponentially positive, then the matrix −L is marginally stable and the system (3) converges to…”

“…Even without using effective resistance, such conditions appear very difficult to obtain, see e.g. [2], [1].…”

Convergence Analysis of Signed Nonlinear Networks

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Bifurcation analysis of two coupled Jansen-Rit neural mass models

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