On the largest eigenvalue of a mixed graph with partial orientation

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.

Automated summary

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.