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<p>The eccentricity matrix <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\epsilon (G),\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> of a connected graph <i>G</i> is obtained by retaining the maximum distance from each row and column of the distance matrix of <i>G</i>, and the other entries are assigned with 0. In this paper, we discuss the eccentricity spectrum of the subdivision vertex (edge) join of regular graphs. Also, we obtain new families of graphs having irreducible or reducible eccentricity matrix. Furthermore, we use these results to construct infinitely many <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\epsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>-cospectral graph pairs as well as infinitely many pairs and triplets of non <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\epsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>-cospectral <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\epsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>-equienergetic graphs. Moreover, we present some new family of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\epsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>-integral graphs.</p>

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On irreducibility of eccentricity matrix of graphs and construction of \(\epsilon \)-equienergetic graphs

  • 期刊论文

  • 发表日期：

    2026年3月9日

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  • Anjitha Ashokan,
  • Chithra A. V.

摘要

The eccentricity matrix \(\epsilon (G),\) ϵ ( G ) , of a connected graph G is obtained by retaining the maximum distance from each row and column of the distance matrix of G, and the other entries are assigned with 0. In this paper, we discuss the eccentricity spectrum of the subdivision vertex (edge) join of regular graphs. Also, we obtain new families of graphs having irreducible or reducible eccentricity matrix. Furthermore, we use these results to construct infinitely many \(\epsilon \) ϵ -cospectral graph pairs as well as infinitely many pairs and triplets of non \(\epsilon \) ϵ -cospectral \(\epsilon \) ϵ -equienergetic graphs. Moreover, we present some new family of \(\epsilon \) ϵ -integral graphs.