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arXiv · 2508.11970

Vertex energy distributions in regular graph structures

H. M. Nagesh·U. Vijaya Chandra Kumar·N. Narahari
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Abstract

The energy of a vertex $v_i$ in a graph $G$ is defined as $\mathcal{E}_G(v_i) = |A|_{ii}$, where $A$ is the adjacency matrix of $G$, $A^*$ denotes the conjugate transpose of $A$, and $|A| = (AA^*)^{1/2}$. The total energy of the graph, $\mathcal{E}(G)$, is then the sum of the energies of all vertices: $\mathcal{E}(G) = \mathcal{E}_G(v_1) + \mathcal{E}_G(v_2) + \dots + \mathcal{E}_G(v_n)$. In this paper, we compute the vertex energy for several well-known regular graphs, including the Frucht graph, Desargues graph, Tutte-Coxeter graph, Heawood graph, Shrikhande graph, and Petersen graph.

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H. M. Nagesh, U. Vijaya Chandra Kumar, N. Narahari. 2025-08-16. Vertex energy distributions in regular graph structures. https://arxiv.org/abs/2508.11970

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