Let G be a finite, simple, undirected graph. The maximum degree matrix M(G) and the minimum degree matrix m(G) assign to each adjacent pair of vertices the maximum and minimum of their degrees, respectively, thereby encoding extremal degree interactions between vertices. In this paper, the spectral and energy properties of these degree-based matrices are examined in a unified framework. A complete spectral characterization of graph regularity via six equivalent conditions on M(G) and m(G) is established, together with results on irreducibility, inertia, spectral radius, diagonal dominance, and energy. Closed-form spectral and energy expressions are derived for complete graphs, wheel graphs, star graphs, paths, and trees. The central result proves that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$E_m(G) < E_M(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <msub> <mi>E</mi> <mi>M</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for every connected non-regular graph, with equality if and only if G is regular; a quantitative lower bound on the energy gap in terms of edge-degree variance is also established. Sharp extremal bounds with unique equality cases are obtained for trees, and tight asymptotic formulas for all standard graph families are proved rigorously. A systematic four-way comparison of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$E_M(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mi>M</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$E_m(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, Laplacian energy LE(G), and eccentricity energy <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$E_{\varepsilon }(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>E</mi> <mi>ε</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> demonstrates that these invariants capture genuinely complementary local and global structural information. These results contribute to the spectral study of degree-based extremal matrices and open several directions for further investigation.

Energy and Spectra of Degree-Based Graph Matrices

期刊论文
发表日期：

2026年7月3日

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Sakunthala Srinivasan,
Janani Rajasekar

摘要

Let G be a finite, simple, undirected graph. The maximum degree matrix M(G) and the minimum degree matrix m(G) assign to each adjacent pair of vertices the maximum and minimum of their degrees, respectively, thereby encoding extremal degree interactions between vertices. In this paper, the spectral and energy properties of these degree-based matrices are examined in a unified framework. A complete spectral characterization of graph regularity via six equivalent conditions on M(G) and m(G) is established, together with results on irreducibility, inertia, spectral radius, diagonal dominance, and energy. Closed-form spectral and energy expressions are derived for complete graphs, wheel graphs, star graphs, paths, and trees. The central result proves that $E_m(G) < E_M(G)$ $E_{m} (G) < E_{M} (G)$ for every connected non-regular graph, with equality if and only if G is regular; a quantitative lower bound on the energy gap in terms of edge-degree variance is also established. Sharp extremal bounds with unique equality cases are obtained for trees, and tight asymptotic formulas for all standard graph families are proved rigorously. A systematic four-way comparison of $E_M(G)$ $E_{M} (G)$ , $E_m(G)$ $E_{m} (G)$ , Laplacian energy LE(G), and eccentricity energy $E_{\varepsilon }(G)$ $E_{ε} (G)$ demonstrates that these invariants capture genuinely complementary local and global structural information. These results contribute to the spectral study of degree-based extremal matrices and open several directions for further investigation.