On the Vertex Seidel Energy of Graphs

Abstract

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.

Figures (3)

• Figure 1: Left: $K_2$ with vertex Seidel energy $\sqrt{1}=1$. Right: Conference type example of order $6$ with vertex Seidel energy $\sqrt{5}$.
• Figure 2: A connected 3-regular graph (modified Petersen) with non-constant vertex Seidel energy.
• Figure 3: Contour in the complex $t$-plane for the Coulson integral. The pole at $t=\mathrm{i}(n-1)$ (upper half-plane) is enclosed and contributes via its residue; the pole at $t=-\mathrm{i}$ lies in the lower half-plane and does not contribute.

Theorems & Definitions (23)

• Definition 1
• Proposition 1
• proof
• Proposition 2
• proof
• Corollary
• proof
• Corollary
• proof
• Corollary