Abstract:This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind introduced by Mohar [21]. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from u to v is equal to the sixth root of unity ω = 1+i √ 3 2 (and its symmetric entry is ω = 1−i √ 32 ); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. The main results of this paper include the following: Some interesting properties ar… Show more
“…Clearly, the Hermitian adjacency matrix of a mixed graph is a natural generalization of the adjacency matrix of a simple graph. Therefore the mixed graph attracts more and more researchers' attention; see [13,21,24,25,28,35,36,52,51,50,59,60].…”
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.
“…Clearly, the Hermitian adjacency matrix of a mixed graph is a natural generalization of the adjacency matrix of a simple graph. Therefore the mixed graph attracts more and more researchers' attention; see [13,21,24,25,28,35,36,52,51,50,59,60].…”
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.
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