Mixed graphs with cut vertices having exactly two positive eigenvalues

Abstract

A mixed graph is obtained by orienting some edges of a simple graph. The positive inertia index of a mixed graph is defined as the number of positive eigenvalues of its Hermitian adjacency matrix, including multiplicities. This matrix was introduced by Liu and Li, independently by Guo and Mohar, in the study of graph energy. Recently, Yuan et al. characterized the mixed graphs with exactly one positive eigenvalue. In this paper, we study the positive inertia indices of mixed graphs and characterize the mixed graphs with cut vertices having positive inertia index 2.

Combinatorial necessary conditions for regular graphs to induce periodic quantum walks

“…This is introduced by Mohar [30]. We refer to [1,9,10,11,26,45] as recent studies on Hermitian adjacency matrices. Note that H η (G ± ) coincides with the ordinary adjacency matrix of G ± .…”