Vertex Seidel Energy of Graphs under Parameterized Twisted Adjacency Matrix

Abstract

We introduce the concept of parameterized twisted Seidel energy by incorporating a real parameter $\alpha$ into the Seidel matrix definition. This generalization yields a family of graph energies that interpolate between classical Seidel energy ($\alpha = 0$) and complementary Seidel energy ($\alpha = 1$). We establish spectral formulas, compute exact values for several graph families, derive sharp bounds with complete proofs, and prove invariance properties under graph operations. A generalized Coulson-type integral representation is obtained, and we characterize graphs attaining extremal values. New results include monotonicity properties, characterization of $\alpha$-equienergetic graphs, and perturbation bounds.