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<p>The spectrum of a graph <i>G</i> is the spectrum of its adjacency matrix and the distance spectrum of <i>G</i> is the spectrum of its distance matrix. A graph is called <i>integral</i> if all of its adjacency eigenvalues are integers and <i>distance</i> <i>integral</i> if all of its distance eigenvalues are integers. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(G=(P_1 \cup P_2,E)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mn>1</mn> </msub> <mo>∪</mo> <msub> <mi>P</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a regular bipartite graph of diameter 3. In the present paper, we show that there is a close relationship between the spectrum of <i>G</i> and its distance spectrum. We show how we can determine the distance spectrum of <i>G</i> when we have its spectrum. In particular, we show that if <i>G</i> is integral then <i>G</i> is distance integral. Also, we determine the distance spectrum of some classes of graphs.</p>

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The distance spectrum of regular bipartite graphs of diameter 3

  • 期刊论文

  • 发表日期：

    2026年3月2日

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  • S. Morteza Mirafzal

摘要

The spectrum of a graph G is the spectrum of its adjacency matrix and the distance spectrum of G is the spectrum of its distance matrix. A graph is called integral if all of its adjacency eigenvalues are integers and distance integral if all of its distance eigenvalues are integers. Let \(G=(P_1 \cup P_2,E)\) G = ( P 1 ∪ P 2 , E ) be a regular bipartite graph of diameter 3. In the present paper, we show that there is a close relationship between the spectrum of G and its distance spectrum. We show how we can determine the distance spectrum of G when we have its spectrum. In particular, we show that if G is integral then G is distance integral. Also, we determine the distance spectrum of some classes of graphs.