Some Mixed Graphs Determined by Their Spectrum

TL;DR

This paper investigates which mixed graphs are uniquely identified by their Hermitian spectrum, focusing on cycles and paths, and classifies their spectral properties and cospectral mates.

Contribution

It characterizes all mixed paths and cycles that are determined by their Hermitian spectrum, expanding understanding of spectral graph determination in mixed graphs.

Findings

All mixed paths of even order except P8 and P14 are DHS.

Mixed paths of odd order except P3 are not DHS.

Certain mixed cycles are DHS, while others are not.

Abstract

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 $ \{v_{1}, \ldots , v_{n}\} $, is the matrix $ H=[h_{ij}]_{n \times n} $, where $ h_{ij}=-h_{ji}=i $ if there is a directed edge from $ v_{i} $ to $ v_{j} $, $ h_{ij}=1 $ if there exists an undirected edge between $v_i$ and $v_{j}$, and $h_{ij}=0$ 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 ($C_{n}$), a mixed cycle with exactly one directed edge ($C_{n}^{1}$), or a mixed cycle with exactly two consecutive directed edges with the same direction ($C_{n}^{2}$) 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 $P_{8}$ and $P_{14}$, are DHS. It is also shown that mixed paths of odd order, except $P_{3}$, are not DHS. Also, all cospectral mates of $P_{8}$, $P_{14}$ and $P_{4k+1}$ and two families of cospectral mates of $P_{4k+3}$, where $k\geq1$, are introduced. Finally, we show that the mixed cycles $C_{2k}$ and $C_{2k}^{2}$, where $k\geq3$, are not DHS, but the mixed cycles $C_{4}$, $C_{4}^{2}$, $C_{2k+1}$, $C_{2k+1}^{2}$, $C_{2k+1}^{1}$ and $C_{2j}^{1}$ except $C_{7}^{1}$, $C_{9}^{1}$, $C_{12}^{1}$ and $C_{15}^{1}$, are DHS, where $k\geq1$ and $j\geq2$.