Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Articles in Press 2025 11 06 Vertex energy invariance in double graphs and bipartite double covers 15039 10.22049/cco.2025.30721.2592 EN Cahit Dede Department of Mathematics, Faculty of Science, Selçuk University, Sel¸cuklu, 42003, Konya, Türkiye 0000-0001-5362-9513 Kalpesh M. Popat Department of Mathematics, Saurashtra University, Rajkot, 360005, Gujarat, India Journal Article 2025 06 15 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. https://comb-opt.azaruniv.ac.ir/article_15039_9086160695eeb8735766a874ce2b332a.pdf