On the Vertex Seidel Energy of Graphs

We introduce the vertex Seidel energy via the diagonal entries of the absolute Seidel matrix. We establish a spectral formula, compute exact values for several graph families, derive bounds, and present a Coulson-type integral representation for analytical study of this invariant. We also show that vertex Seidel energy is invariant under Seidel switching and complementation.

💡 Research Summary

The paper introduces a new vertex‑based invariant for simple undirected graphs called the vertex Seidel energy. Starting from the Seidel matrix (S(G)=J-I-2A(G)), the authors consider its absolute value (|S|=(S^{2})^{1/2}) and define the energy of a vertex (v_i) as the diagonal entry (|S|_{ii}). In spectral terms this can be written as
\