Journal
LINEAR ALGEBRA AND ITS APPLICATIONS
Volume 627, Issue -, Pages 150-161Publisher
ELSEVIER SCIENCE INC
DOI: 10.1016/j.laa.2021.06.003
Keywords
Mixed graph; Hermitian adjacency matrix; Largest eigenvalue; Interlacing family; Matching polynomial; Partial orientation
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Funding
- National Natural Science Foundation of China [11871073, 11771016]
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The paper proves the existence of a partial orientation sigma with respect to T for a connected graph G, such that the largest eigenvalue of the resulting graph's Hermitian adjacency matrix satisfies a specific condition.
Let G be a connected graph and let T be an acyclic set of edges of G. A partial orientation sigma of G with respect to T is an orientation of the edges of G except those edges of T, the resulting graph associated with which is denoted by G(T)(sigma). In this paper we prove that there exists a partial orientation sigma of G with respect to T such that the largest eigenvalue of the Hermitian adjacency matrix of G(T)(sigma) is at most the largest absolute value of the roots of the matching polynomial of G. (C) 2021 Elsevier Inc. All rights reserved.
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