On the Energy and Nullity of Uniform Semigraphs
DOI:
https://doi.org/10.30495/jme.v20i0.3471Abstract
The energy of the semigraph is defined as the sum of the absolute eigenvalues of its adjacency matrix. The study of semigraph energy provides a valuable framework for analyzing the stability, connectivity, and structural properties of hyper-networks.
The nullity of a graph encodes both algebraic and structural information.
We extend this concept to semigraphs, an ordered-tuple generalisation of graphs, and derive closed-form formulas for the nullity and energy of
(i) Path-Uniform Semigraphs $P_{k+1}^{\,n}$,
(ii) Cycle-Uniform Semigraphs $C_{k}^{\,n}$,
(iii) Star-type Semigraphs $S_{2,n}^{3}$ and their $3$-Uniform counterparts $S_{n}^{3}$ and
(iv) Rooted $3$-Uniform Semigraph trees $T_{n}^{3}$,
Sharp lower and upper bounds are established in terms of pure middle and isolated vertices, showing that every non-trivial semigraph has nullity at most $n-2$. Beyond enriching semigraph theory with spectral techniques, our results supply new tools for analysing controllability, energy, and redundancy in higher-order network models.
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