<p>The skew spectrum of a bipartite oriented graph is completely determined by the spectrum of the adjacency matrix of a related signed graph, and in the bipartite case this relation is particularly intuitive. In this paper, we determine the conditions under which an oriented graph <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$G'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>G</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> yields a bipartite oriented line graph <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {L}(G')$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, construct the associated signed graph, and compute the skew spectrum of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {L}(G')$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We also examine two specific cases: when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {L}(G')$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> can be switched to a canonical orientation, and when it forms a forest. Furthermore, we establish several structural properties of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {L}(G')$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, including a Krausz-type covering characterisation.</p>

Some Spectral Properties of Oriented Line Graphs

期刊论文
发表日期：

2025年10月2日

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Zoran Stanić

摘要

The skew spectrum of a bipartite oriented graph is completely determined by the spectrum of the adjacency matrix of a related signed graph, and in the bipartite case this relation is particularly intuitive. In this paper, we determine the conditions under which an oriented graph $G'$ $G^{'}$ yields a bipartite oriented line graph $\mathcal {L}(G')$ $L (G^{'})$ , construct the associated signed graph, and compute the skew spectrum of $\mathcal {L}(G')$ $L (G^{'})$ . We also examine two specific cases: when $\mathcal {L}(G')$ $L (G^{'})$ can be switched to a canonical orientation, and when it forms a forest. Furthermore, we establish several structural properties of $\mathcal {L}(G')$ $L (G^{'})$ , including a Krausz-type covering characterisation.