Characterizing signed mixed graphs with small eigenvalues

摘要

A mixed graph M-G is obtained from a simple graph G by orienting an edge subset of G. A signed mixed graph is a mixed graph with arcs and edges signed + or -. The unit Eisenstein matrix (epsilon-matrix for short) of a signed mixed graph was recently introduced by Wissing and van Dam [32]. This novel matrix is indexed by the vertices of the signed mixed graph, and the entry corresponding to a positive arc from u to v is equal to omega = 1+i root 3/2 (and its symmetric entry is (omega) over bar = 1-i root 3/2); the entry corresponding to a negative arc is equal to -omega (and its symmetric entry is -(omega) over bar); the entry corresponding to a positive edge is equal to 1; the entry corresponding to a negative edge is equal to -1; and 0 otherwise. In this paper, we study the spectral properties of this epsilon-matrix. We characterize all the signed mixed graphs whose eigenvalues are contained in (-alpha, alpha) for alpha is an element of {root 2, root 3, root 2} .

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