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Mixed paths and cycles determined by their spectrum

Akbari, S ; Sharif University of Technology | 2020

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  1. Type of Document: Article
  2. DOI: 10.1016/j.laa.2019.09.014
  3. Publisher: Elsevier Inc , 2020
  4. Abstract:
  5. A mixed graph is obtained from a graph by orienting some of its edges. The Hermitian adjacency matrix of a mixed graph with the vertex set {v1,…,vn}, is the matrix H=[hij]n×n, where hij=−hji=i if there is a directed edge from vi to vj, hij=1 if there exists an undirected edge between vi and vj, and hij=0 otherwise. The Hermitian spectrum of a mixed graph is defined to be the spectrum of its Hermitian adjacency matrix. In this paper we study mixed graphs which are determined by their Hermitian spectrum (DHS). First, we show that each mixed cycle is switching equivalent to either a mixed cycle with no directed edges (Cn), a mixed cycle with exactly one directed edge (Cn 1), or a mixed cycle with exactly two consecutive directed edges with the same direction (Cn 2) and we determine the spectrum of these three types of cycles. Next, we characterize all DHS mixed paths and mixed cycles. We show that all mixed paths of even order, except P8 and P14, are DHS. It is also shown that mixed paths of odd order, except P3, are not DHS. Also, all cospectral mates of P8, P14 and P4k+1 and two families of cospectral mates of P4k+3, where k≥1, are introduced. Finally, we show that the mixed cycles C2k and C2k 2, where k≥3, are not DHS, but the mixed cycles C4, C4 2, C2k+1, C2k+1 2, C2k+1 1 and C2j 1 except C7 1, C9 1, C12 1 and C15 1, are DHS, where k≥1 and j≥2. © 2019 Elsevier Inc
  6. Keywords:
  7. Hermitian spectrum ; Mixed graphs ; Linear algebra ; Mathematical techniques ; Adjacency matrices ; Cycle ; Directed edges ; Even orders ; Hermitians ; Mixed graph ; Path ; Vertex set ; Directed graphs
  8. Source: Linear Algebra and Its Applications ; Volume 586 , 2020 , Pages 325-346
  9. URL: https://www.sciencedirect.com/science/article/abs/pii/S0024379519303970