Vertex energy invariance in double graphs and bipartite double covers

Articles in Press

Document Type : Original paper

Authors

1 Department of Mathematics, Faculty of Science, Selçuk University, Sel¸cuklu, 42003, Konya, Türkiye

2 Department of Mathematics, Saurashtra University, Rajkot, 360005, Gujarat, India

Abstract

Vertex energy is a local spectral invariant that measures the contribution of individual vertices to the overall energy of a graph. Understanding how vertex energy behaves under graph transformations is essential for both theoretical insights and practical applications in spectral graph theory and network analysis. In this paper, we investigate the preservation of vertex energy under two fundamental graph constructions: the double graph and the bipartite double cover. We prove that for any connected graph \( G \), the vertex energies of the duplicated vertices in both \( \mathrm{D}(G) \) and \( \mathrm{DC}(G) \) remain identical to those in \( G \). These results demonstrate the robustness of vertex energy as a spectral measure invariant under these duplication operations. To illustrate the theorems, we provide explicit examples and computational verifications using SAGEMATH, with code publicly available for reproducibility. Our findings contribute to the deeper understanding of spectral properties in graph operations and open avenues for further research in spectral invariants of graph transformations.

Keywords

Main Subjects

References

[1] O. Arizmendi, J.F. Hidalgo, and O. Juarez-Romero, Energy of a vertex, Linear Algebra Appl. 557 (2018), 464–495.
https://doi.org/10.1016/j.laa.2018.08.014
[2] O. Arizmendi, B.C. Luna-Olivera, and M.R. Ibáñez, Coulson integral formula for the vertex energy of a graph, Linear Algebra Appl. 580 (2019), 166–183. https://doi.org/10.1016/j.laa.2019.06.009
[3] O. Arizmendi and S. Sigarreta, The change of vertex energy when joining trees, Linear Algebra Appl. 687 (2024), 117–131. https://doi.org/10.1016/j.laa.2024.01.014
[4] D.M. Cvetković, P. Rowlinson, and S. Simić, An introduction to the theory of graph spectra, vol. 75, Cambridge university press Cambridge, 2010.
[5] H.A. Ganie, S. Pirzada, and A. Iványi, Energy, Laplacian energy of double graphs and new families of equienergetic graphs, Acta Univ. Sapientiae, Informatica 6 (2014), no. 1, 89–116.
[6] I. Gutman, The energy of a graph, Ber. Math. Statist. Sekt. Forsch-ungszentram Graz. 103 (1978), 1–22.
[7] I. Gutman and B. Furtula, Vertex energies of graphs- A tutorial, MATCH Commun. Math. Comput. Chem. 93 (2025), no. 3, 691–698. https://doi.org/10.46793/match.93-3.691G
[8] X. Li, Y. Shi, and I. Gutman, Graph Energy, Springer, New York, 2012.
[9] V. Nikiforov, The energy of graphs and matrices, J. Math. Anal. Appl. 326 (2007), no. 2, 1472–1475. https://doi.org/10.1016/j.jmaa.2006.03.072
[10] H.S. Ramane, S.Y. Chowri, T. Shivaprasad, and I. Gutman, Energy of vertices of subdivision graphs, MATCH Commun. Math. Comput. Chem. 93 (2025), no. 3, 701–711. https://doi.org/10.46793/match.93–3.701R
[11] The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 9.1.0), 2021, https://www.sagemath.org.
[12] S.K. Vaidya and K.M. Popat, Some new results on energy of graphs, MATCH commun. Math. Comput. Chem. 77 (2017), no. 4, 589–594.
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 06 November 2025

Files

Share

How to cite

Statistics