Discrete Topological Index

• Discrete topological index is a class of mathematical invariants that classify the structure of discrete objects using algebraic, combinatorial, and operator-theoretic methods.
• They are applied to detect phase transitions, classify symmetry-protected states, and analyze properties in quantum spin chains, chemical graphs, dynamical systems, and digital image processing.
• Their robustness under perturbations and symmetry-preserving deformations makes these indices crucial for both practical computational algorithms and theoretical classifications.

Discrete topological index refers to a class of mathematical invariants that quantify or classify the topological structure of discrete objects—such as graphs, finite dynamical systems, spin chains, digital images, or lattice models—by means of algebraic, combinatorial, or operator-theoretic constructs. These indices appear across topology, mathematical physics, chemical graph theory, combinatorics, and computational dynamics, often serving as discrete analogues or extensions of classical topological invariants (e.g., the Euler characteristic, the index in Atiyah–Singer theory) and play a crucial role in classifying phases, detecting symmetry-protected topology, and guiding computational algorithms.

1. Discrete Spin Chain and Fermionic $\mathbb{Z}_2$ Indices

Discrete topological indices in quantum lattice systems serve as phase descriptors immune to local perturbations and symmetry-preserving deformations. For infinite one-dimensional $S=1$ spin chains with short-range Hamiltonians respecting U(1) and a “protecting” symmetry (either $\pi$ rotation about $x$, reflection, or time-reversal), Tasaki constructs a rigorous discrete $\mathbb{Z}_2$ index for ground states, differentiating between the topologically nontrivial Haldane phase (AKLT) and trivial paramagnets (Tasaki, 2018). The index is computed as the sign of the expectation of the Affleck–Lieb twist operator in the thermodynamic limit: $$\sigma(H) = \lim_{\ell\to\infty} \mathrm{sign}\,\omega(V_\ell) \in \{+1,-1\}$$ This sign is invariant under any smooth, gap-preserving deformation of the Hamiltonian that respects the underlying symmetries. In the AKLT chain, $\sigma_{\mathrm{AKLT}}=-1$, while for product states, $\sigma_{\mathrm{trivial}}=+1$, and the sign flips only at a true topological phase transition.

Analogously, for free-fermion models in arbitrary dimension, a noncommutative $\mathbb{Z}_2$-projection index (and its $\mathbb{Z}$ generalizations) is constructed via operator algebras and Fredholm theory. The index is stable under homotopies, detects phase transitions (gap closings), and, in disordered settings, provides robust classification even beyond translation-invariant systems (Aza et al., 2020, Katsura et al., 2016).

2. Graph-Theoretic Discrete Topological Indices

In discrete mathematics and chemical graph theory, topological indices are graph invariants computed (in various manners) from the structure of $G=(V,E)$. Degree-based indices (e.g., Zagreb indices, F-index, Hyper-Zagreb index) are concrete polynomial or product expressions in vertex degrees and have closed-form recursions for subdivision or inflation sequences: $$M_1(G) = \sum_{v\in V} d_G(v)^2, \quad F(G) = \sum_{v\in V} d_G(v)^3$$ with generalizations for higher subdivisions/semi-total-point graphs (De, 2017), and their growth rates controlled combinatorially.

Distance-based indices, such as the Wiener index $W(G)$ and the resolving topological index $\mathcal{R}(G)$, probe the metric structure. The resolving index, based on the concept of a “resolving share” for pairs of vertices, quantifies the extent to which each vertex distinguishes other pairs: $$r_w(u,v) = \begin{cases} 1/|R(u,v)|, & \text{if } w \in R(u,v) \ 0, & \text{otherwise} \end{cases} ,\quad \mathcal{R}(G) = \sum_{w\in V} \left( \frac{1}{|R(w)|} \sum_{(u,v)\in R(w)} r_w(u,v) \right)$$ This index complements the metric dimension and captures geometric and chemical features unaddressed by purely degree-based descriptors (Salman et al., 2014).

3. Discrete Conley Index in Dynamical Systems

The Conley index for discrete dynamical systems generalizes topological fixed-point invariants to maps or multivalued correspondences on finite spaces or compact metric spaces. It is defined via index pairs $(N,L)$ for isolated invariant sets and an induced quotient map $f_{(N,L)}: N/L \to N/L$. For classical discrete flows, the index is the homotopy class [shift-equivalence class] of $f_{(N,L)}$ (Weilandt, 2018). Notably, the mapping-torus interpretation refines comparison, with the mapping torus $T(f_{(N,L)})$ yielding a homotopy-type invariant that carries the full spectrum of algebraic invariants (homology, $\pi_1$, etc.) and is stable under continuation.

For finite topological spaces, the index is constructed by order-theoretic methods involving lower-semicontinuous multivalued maps, isolating neighborhoods, and induced maps in homology, equipped with a normal functor (e.g., Leray reduction) that ensures invariance and continuation properties (Barmak et al., 2023).

4. Digital Topology and Local Indices in Discrete Images

In digital image analysis, local discrete topological indices are computed via “topological numbers” in $2$D (or higher) grids. By counting local adjacency relations and connected components in the $3\times3$ neighborhood, six point types are classified (isolated, interior, simple, curve, 3-junction, 4-junction), using $(T_n(x,X), T_{\bar n}(x, \bar X))$ (Lohou, 8 Jan 2025). These assignments preserve global invariants such as Betti numbers and Euler characteristic, as the removal of a “simple point” leaves both object and background connectivity unchanged.

5. Index Theory for Discrete Groups and Operator Algebras

Discrete topological index theory appears in higher-index constructions for countable groups and noncommutative spaces, notably in the context of the Baum–Connes conjecture and C*-algebraic K-theory. Geometric cycles for $K_*^{geom}(\Gamma)$ provide a model for assembly maps, with the Chern–Baum–Connes assembly morphism producing explicit index-pairing formulas in terms of integrals over fixed-point sets and delocalized Todd/Chern forms: $$\langle \Phi([M,x]), \tau \rangle = \sum_{\langle g \rangle} \int_{M_g} ch_M^g(x) \wedge Td_g^M \wedge \pi_g^*(\tau_g)$$ This formula generalizes the classical fixed-point formula to arbitrary discrete groups and fully incorporates the cyclic periodic cohomology of the group algebra (Rouse et al., 2020).

Operator-algebraic treatments of spatial interfaces and domain walls on discrete lattices (e.g., 1D SSH chains) yield discrete topological indices via KK-theory, connecting bulk topological invariants to interface states through exact sequences and boundary maps. The index assigns robustness and classifies zero modes (Bourne, 25 Aug 2025).

6. Discrete Topological Indices in Computational and Morse Theory

Discrete Morse theory encodes topological invariants for combinatorial complexes via discrete Morse index, combinatorial Poincaré indices, and gradient vector fields. For a cell complex, critical cells are assigned index $(-1)^k$ (where $k$ is the dimension), and the alternating sum recovers the Euler characteristic: $$\sum_{\sigma \in \mathrm{Crit}(f)} (-1)^{\mathrm{ind}(\sigma)} = \chi(K)$$ Algorithms for computing these indices operate in polynomial or linear time depending on data structures, and the invariants are stable under subdivision and elementary collapses (Prishlyak, 1 Feb 2025).

7. Robustness, Invariance, and Applications

Discrete topological indices manifest universal invariance under homeomorphism, local perturbations, and symmetry-preserving deformations, paralleling their continuous counterparts. They serve as sharp tools for:

• Classification of symmetry-protected topological phases in quantum systems
• Structure–property mapping in chemistry via molecular graph invariants
• Topological recognition and preprocessing in image analysis and digital topology
• Computation and reduction in dynamical systems and combinatorial topology
• Fundamental index-pairings in K-theory, operator algebras, and group cohomology

Research continues on extending index concepts to multivalued maps, higher dimensions, spectral and noncommutative settings, and algorithmic frameworks for real-world data.

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Table: Summary of Selected Discrete Topological Indices

Index Type	Context	Invariant Value(s)
$\mathbb{Z}_2$ twist sign (Tasaki)	$S=1$ spin chains, SPT classification	$\sigma(H)\in\{+1,-1\}$
Resolving topological index	Molecular/chemical graph theory	$\mathcal{R}(G)\in (0, |G|/2]$
Mapping torus Conley index	Discrete dynamical systems	Homotopy type of $T(f_{(N,L)})$
Local topological number ($T_n$)	Digital images, topology preservation	$(0,1), (1,0), \ldots, (4,4)$
Discrete Morse index	Cell complexes, dynamical systems	$(-1)^k$, Euler characteristic
Operator K-theory index	Interfaces/domain walls (lattice, bulk)	$K$-theory class, winding difference

These indices encode discrete analogues of classical topology and reveal new phenomena arising from the combinatorial, algebraic, and computational structure of non-continuous systems.