Sigma Index across Disciplines

• Sigma Index is a domain-dependent metric that measures structural irregularity, such as squared degree differences in graphs and concentration profiles in galaxies.
• In quantum field theory and geometric analysis, it serves as a tool for diagnosing integrability and energy stability through analytic and operator formulations.
• In number theory, the abundancy index (Sigma Index) quantifies divisor richness, linking perfect numbers, superabundant sequences, and conditions like Robin’s inequality.

The Sigma Index is a domain-dependent quantitative metric whose mathematical definitions, physical interpretations, and applications vary across graph theory, extragalactic astronomy, quantum field theory, analytic number theory, and geometric analysis. It typically encodes the irregularity, concentration, or stability of structures or systems—frequently via quadratic sum, partitioning, or index-theoretic formulations. Below, the major usages are classified and detailed according to the arXiv record, focusing on irregularity indices in graphs, the Sérsic index in galaxy structure, the integrability index in quantum sigma models, and the abundancy index in number theory.

1. Sigma Index in Graphs and Irregularity Quantification

The primary graph-theoretic definition is the sigma irregularity index, measuring the degree disparity across edges: $$\sigma(G) = \sum_{uv \in E(G)} ( \deg(u) - \deg(v) )^2$$ This index, originally formulated by Gutman et al., is central to the paper of graph irregularity, with a quadratic amplifying effect over the well-known Albertson index, $$\operatorname{irr}(G) = \sum_{uv \in E(G)} | \deg(u) - \deg(v) |$$ (Hamoud et al., 22 Oct 2025).

Key Results and Structural Applications:

• Extremal Bounds: Given a tree $T$ of order %%%%2%%%%, Gutman et al. prove

$$\sigma_{\max}(T) = (n-1)(n-2), \quad \sigma_{\min}(T) = 0$$

The maximum is unique to the star configuration (one vertex of degree $n-1$, others degree 1); the minimum occurs for regular graphs (Hamoud et al., 22 Oct 2025).

• Prescribed Degree Sequences: For trees and caterpillar graphs with degree sequence $\mathscr{D} = (d_1, d_2, \ldots, d_n)$ (non-increasing), the sigma index is calculated by summing squared differences between adjacent degrees along backbone and leaf connectors (Hamoud et al., 8 May 2024). Closed formulas for caterpillar trees $\mathscr{C}(n,m)$ with $n$ backbone and $m$ pendants per vertex include

$$\sigma(\mathscr{C}(n,m)) = \begin{cases} 2m^3 & \text{if}\ n=2 \ 2m^3 + m - 2 & \text{if}\ n \geq 3 \end{cases}$$

(Hamoud et al., 22 Oct 2025).

• Upper/Lower Bounds: In extremal graph theory, sharp analytical bounds are developed by leveraging degree sequence statistics (max, min, consecutive differences, means, etc.). For trees with non-decreasing sequence $\mathscr{D} = (d_1, \ldots, d_n)$ and related arithmetic mean and local difference sequences $\mathscr{A}$, $\mathscr{R}$, the main results are

$$\sigma(T) \geq \operatorname{irr}(T) + \left\lfloor \frac{n-2}{a_r - t_m} \right\rfloor + \Delta (\Delta_{\mathscr{A}} - \Delta_{\mathscr{R}})^2$$

$$\sigma_{\max}(T) \leq \frac{1}{2(\Delta - 3)} \left( \left\lfloor \frac{3n^2}{4} \right\rfloor \left\lceil \frac{n^2}{4} \right\rceil \right)$$

with $\Delta$ as maximum degree, $a_r$ largest mean in $\mathscr{A}$, $t_m$ largest consecutive half-difference in $\mathscr{R}$ (Hamoud et al., 23 Sep 2025).

Connections to Classical Indices:

There are precise relations between the sigma index, the second Zagreb index ($M_2(G) = \sum_{uv \in E(G)} \deg(u)\deg(v)$), and the forgotten topological index ($F(G)$): $\sigma(G) = F(G) - 2M_2(G)$ (Hamoud et al., 8 Oct 2025). Such relationships clarify the network of classical molecular descriptors in QSPR/QSAR and network analysis.

Role in Structured Graph Classes:

For k-cyclic graphs (order $n$, size $n + k - 1$), the maximum sigma index is uniquely attained by $H_{n,k}$—a star graph with $k$ additional edges among pendants. The explicit bound is

$\sigma(G) \leq (n-1)(n-2)^2 - 2k(2n-5) + k^2(k-1)$

(Ali et al., 2022). This result generalizes previous findings on maximum irregularity in connected graph families.

2. Sigma Index (Sérsic Index) in Galaxy Structure

In extragalactic astronomy, the Sérsic index—sometimes informally dubbed the 'sigma index'—quantifies the central concentration of galaxy surface brightness profiles via a Sérsic model fit: $$I(r) \propto \exp\left[ -b_n \left( \frac{r}{r_e} \right)^{1/n} \right ]$$ with index $n$ describing the profile shape (Kelvin et al., 2011).

Multi-Wavelength Structural Trends:

Analysis of $167,600$ galaxies in SDSS+UKIDSS imaging from the GAMA survey demonstrates:

• Early-type galaxies (ETGs): $n_K \simeq 3.63$, $n_g \simeq 2.79$, with a $\sim$30% increase in $n$ from $g$ to $K$ band.
• Late-type galaxies (LTGs): $n_K \simeq 1.40$, $n_g \simeq 0.92$, a $\sim$52% increase. This steeper dependency relates to dust attenuation and stellar population gradients.
• Classification Rule: ETGs and LTGs are separated in $K$-band $n$ vs. rest-frame $(u - r)$ color by

$(u - r)_{\mathrm{rest}} = -0.59 \log n_K + 2.07$

(Kelvin et al., 2011).

Structural Implications:

The bandpass dependence of the sigma/Sérsic index underlines that galaxy concentration parameters are not universal, but encode astrophysical phenomena including dust, age gradients, and secular evolution. Near-infrared $n_K$ provides a more robust morphology proxy than optical indices.

3. Sigma Index as Quantum Integrability Diagnostic

In quantum field theory, the integrability index—termed the 'sigma index' in (Komatsu et al., 2019)—provides a rigorous measure of quantum integrability by counting higher-spin conserved currents in two-dimensional sigma models: $\mathcal{I}(J) = c(J,J) - c(J+1,J-1)$ where $c(J,J)$ is the count of conformal primaries of spin $J$, and $c(J+1,J-1)$ counts anomaly (obstruction) operators.

Implications:

• $\mathcal{I}(J) > 0$ guarantees quantum conserved currents of spin $J$, sufficient for integrability.
• In the $O(N)$ model: positive indices at spins 4 and 6, establishing new quantum currents.
• In $\mathbb{CP}^{N-1}$ ($N > 2$) and flag models $U(N)/U(1)^N$: indices negative or non-positive, consistent with non-integrability.

Computational Framework:

Operators are enumerated via gauge-invariant generating functions, plethystic exponentials, and Haar measure projection. The methodology is algorithmic: $$Z(q,x) = \int d\mu_H(y) \exp\left[ \sum_{m=1}^\infty \frac{1}{m} f(q^m,x^m,y^m) \right ]$$ The sigma index is then extracted via expansions in $q$ and $x$.

4. Sigma Index in Analytic Number Theory

The abundancy index (sometimes labeled the 'Sigma Index') is defined as

$I(n) = \frac{\sigma(n)}{n}$

where $\sigma(n)$ is the sum-of-divisors function (Bishop et al., 2021, Guha et al., 2021). This index quantifies the 'divisor richness' of $n$.

Number-Theoretic Applications:

• Perfect numbers: $I(n) = 2$ ($\sigma(n) = 2n$)
• Superabundant numbers: $n$ s.t. $I(m) < I(n)$ for all $m < n$; sequence is infinite.
• Feebly amicable pairs: $(m,n)$ with

$\frac{1}{I(n)} + \frac{1}{I(m)} = 1$

This generalizes amicability to a balancing condition on the reciprocals of abundancy indices (Bishop et al., 2021).

• Connection to Riemann Hypothesis: Robin’s inequality,

$$I(n) < e^{\gamma} \log \log n \quad \Leftrightarrow \text{RH for } n \geq 5041$$

violations must occur at superabundant $n$ (Guha et al., 2021).

Properties:

• $I(n)$ is multiplicative: $I(mn) = I(m) I(n)$ if $(m,n)=1$.
• The set $\{ I(n) : n \geq 2 \}$ is dense in $(1, \infty)$ but has missing rationals ('abundancy outlaws').
• In feebly amicable pairs, not all $n$ admit partners; explicit results are proved (e.g., values $k/(k-m)$ can be isolated).

5. Sigma Index in Analytic and Geometric Index Theory

In global analysis and gauge theory, the Sigma Index may refer to an analytic index morphism of families of elliptic boundary problems on moduli spaces, generalizing the Atiyah-Bott index (Loizides, 2023): $${}^k_T : K^0_T(\_ \Sigma) \rightarrow R^{-\infty}(T)$$ This morphism computes pushforwards of K-theory classes of Fredholm operator families on symplectic quotients. Explicit cohomological formulas, e.g.,

$${}^k_G( \exp(t E^{\Sigma}V) ) = \sum_{g \in T_\ell^/W} \left( \frac{(-1)^{|R_+|} J(g_t)^2}{|T_\ell| \det(1 - t\ell^{-1}H_V(g_t))} \right)^{1-} \delta_{g_t}$$

are employed to generalize the Verlinde formula for moduli of flat connections with boundary (Loizides, 2023).

6. Sigma Index as Energy Index in Harmonic Map Theory

In differential geometry, the Sigma Index is the energy index (count of negative modes) of the second variation of the energy of the harmonic Gauss map from a CMC surface $\Sigma$ (in a Lie group $\mathbb{G}^3$) to the unit sphere in the Lie algebra (Carvalho et al., 14 Jun 2024): $$\operatorname{Ind}_\mathcal{E}( \mathcal{N} ) \geq g(\Sigma)$$ where $g(\Sigma)$ is the genus. For noncompact surfaces, this is generalized to

$$\operatorname{Ind}_\mathcal{E}( \mathcal{N} ) \geq \frac{1}{2} \dim \mathcal{H}^1(\Sigma)$$

where $\mathcal{H}^1(\Sigma)$ is the space of $L^2$ harmonic 1-forms.

Methodological innovations involve harmonic vector field test variations, bundle isometry between tangent and the Gauss map pullback, and analytic localizations.

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In summary, the Sigma Index encompasses a suite of mathematically rigorous measures: in graph theory as a quadratic irregularity index; in astronomy as the Sérsic concentration index; in quantum field theory as an integrability diagnostic; in number theory as a measure of divisor abundance and a tool for generalizing amicability; in global analysis as an analytic K-theory index; and in geometry as a stability index for harmonic maps. Its cross-disciplinary utility is established by closed-form expressions, extremal bounds, and direct links to underlying group-theoretic, topological, and analytic structures.