Some New Results on Seidel Equienergetic Graphs

Abstract: The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Some variants of energy can also be found in the literature which are defined on the concepts of Laplacian matrix, Distance matrix, Common neighbourhood matrix and Seidel matrix. The Seidel matrix of the graph $G$ is the square matrix in which $ij^{th}$ entry is $-1$ or $1$, if the vertices $v_i$ and $v_j$ are adjacent or non-adjacent respectively, and is $0$ , if $v_i=v_j.$ The Seidel energy of $G$ is the sum of the absolute values of the eigenvalues of its Seidel matrix. We present here some graph families which are Seidel equienergetic.

• The paper introduces novel construction techniques using Kronecker products and loop modifications to build Seidel equienergetic, non-cospectral graph pairs.
• The paper proves that equienergeticity is achieved if and only if the base graph has equal numbers of positive and negative Seidel eigenvalues.
• The paper’s findings advance spectral graph theory by providing a systematic framework for generating infinite families of graphs with applications in chemistry and optimization.

Summary of Results on Seidel Equienergetic Graphs

Introduction to Seidel Energy

The paper investigates the structure and construction of Seidel equienergetic graphs, extending foundational work on graph energy, first formalized as the sum of absolute values of eigenvalues of a graph's adjacency matrix. Distinguished from standard energy, Seidel energy arises from the Seidel matrix $S(G)$, where off-diagonal entries are -1 for adjacent vertices and +1 for non-adjacent ones. Seidel energy is defined as $SE(G) = \sum_{i=1}^n |\theta_i|$ for Seidel eigenvalues $\theta_i$.

While Seidel cospectrality implies Seidel equienergeticity, the focus of this work is graphs of same order which are not Seidel cospectral yet Seidel equienergetic, a non-trivial extension; this is motivated by analogous constructions in classical equienergetic graphs but now transferred to the Seidel spectral context.

Construction Techniques for Seidel Equienergetic Graphs

The principal constructions in this paper use the Kronecker product and the modification of graphs via the addition or removal of loops on each vertex. Specifically, two novel families are introduced:

• $D_m(G)$ and $D^*(G)$: For any simple graph $G$, $D_m(G) = K_m \times G$ (where $K_m$ is the complete graph with loops) and $D^*(G) = (K_m \times G_r)_{ur}$ (here $G_r$ denotes $SE(G) = \sum_{i=1}^n |\theta_i|$0 with removed loops). Both constructions yield graphs of order $SE(G) = \sum_{i=1}^n |\theta_i|$1.

Spectral analysis via Kronecker products, leveraging Proposition 1, enables explicit characterization of Seidel eigenvalues for these constructions:

• For $SE(G) = \sum_{i=1}^n |\theta_i|$2: Seidel spectrum is $SE(G) = \sum_{i=1}^n |\theta_i|$3.
• For $SE(G) = \sum_{i=1}^n |\theta_i|$4: Seidel spectrum is $SE(G) = \sum_{i=1}^n |\theta_i|$5.

The paper demonstrates that these two families can produce pairs of non-cospectral graphs with identical Seidel energy, provided the parent graph $SE(G) = \sum_{i=1}^n |\theta_i|$6 has equal numbers of positive and negative Seidel eigenvalues.

Characterization Theorems

Two main theorems rigorously identify the conditions under which the newly constructed pairs are Seidel equienergetic and non-cospectral:

• Theorem 2: $SE(G) = \sum_{i=1}^n |\theta_i|$7 and $SE(G) = \sum_{i=1}^n |\theta_i|$8 are Seidel non-cospectral and equienergetic if and only if $SE(G) = \sum_{i=1}^n |\theta_i|$9 has equal numbers of positive and negative Seidel eigenvalues, assuming $\theta_i$0 for all $\theta_i$1.
• Theorem 3: For iterated constructions $\theta_i$2 and $\theta_i$3, these are Seidel non-cospectral, equienergetic under a stronger restriction $\theta_i$4, with the same balance condition on positive and negative Seidel eigenvalues.

The proofs analyze the sum of absolute values of the new spectral forms, leveraging the symmetry in the eigenvalue distributions and the foundational properties of the Kronecker product.

Theoretical and Practical Implications

These results advance the theoretical understanding of graph energy variants by generalizing equienergetic constructions to Seidel energy. The explicit use of Kronecker products and graph modifications (loops) yields a systematic method for generating infinite families of Seidel equienergetic graphs, including high-order non-cospectral cases.

Practical implications include potential applications in spectral graph theory, chemistry (where graph energy correlates with molecular stability), and combinatorial optimization where spectral properties influence problem hardness. The systematic construction methods may facilitate automated generation of benchmark graphs for energy-based spectral partitioning and other algorithms sensitive to spectral symmetry.

On the theoretical front, the explicit characterization of spectral transformations under Kronecker and loop modifications, and the linkage to eigenvalue balance conditions, provides new tools for an algebraic approach to graph spectral engineering. This sheds light on the constraints necessary for equienergetic families and may motivate further classification of graphs based on Seidel eigenvalue distributions.

Numerical Results and Claims

The strongest claim is the necessary and sufficient condition for non-cospectral Seidel equienergetic pairs: existence hinges exclusively on equal counts of positive and negative Seidel eigenvalues in the base graph.

No explicit numerical tables are included, but the results imply that for all graphs meeting these criteria, the Seidel energies of $\theta_i$5 and $\theta_i$6 (and their iterated forms) coincide exactly; the spectra are distinct but the sum of absolute values of the eigenvalues is provably equal, via algebraic manipulation.

Future Directions

The framework established opens avenues for:

• Classification of all simple graphs with balanced Seidel eigenvalue distributions.
• Generalization to directed graphs, weighted Seidel matrices, and hypergraphs.
• Investigation of Seidel energy preservation under more complex graph products.
• Potential adaptations for applications in quantum chemistry, network theory, and design of spectral-robust networks.

Algorithmic exploration to efficiently recognize and construct large Seidel equienergetic families from arbitrary base graphs may also be of interest.

Conclusion

The paper rigorously extends the theory of Seidel equienergetic graphs, providing constructive methods and explicit spectral characterizations via Kronecker products and loop modifications. The derived conditions for Seidel equienergeticity are both necessary and sufficient under specified eigenvalue constraints, anchoring future directions for algebraic graph spectral analysis and applications where spectral energy is manipulated or preserved.